convergence results for rearrangements: old and new. · pdf fileabstract convergence results...

Convergence Results for Rearrangements: Old and New.

by

Marc Fortier

A thesis submitted in conformity with the requirementsfor the degree of Master of Science

Graduate Department of MathematicsUniversity of Toronto

Copyright © 2010 by Marc Fortier

Abstract

Convergence Results for Rearrangements: Old and New.

Marc Fortier

Master of Science

Graduate Department of Mathematics

University of Toronto

2010

The purpose of this thesis is twofold. On the one hand, it aims to give a thorough

review and exposition of current best results regarding approximating the symmetric

decreasing rearrangement by polarizations and Steiner symmetrizations. These results

include those of Van Schaftingen on explicit universal approximation to the symmetric

decreasing rearrangement by sequences of polarizations as well as his results on almost

sure convergence of rearrangements to the symmetric decreasing rearrangement. They

also include those of Klartag and Milman which yield rates of convergence for Steiner

symmetrizations of convex bodies. On the other hand, new results are proven. We

extend Van Schaftingen’s results on almost sure convergence of polarizations and Steiner

symmetrizations by showing that the conditions on the random variables can be weakened

without affecting almost sure convergence to the symmetric decreasing rearrangement.

Lastly, we derive rates of convergence for polarizations and Steiner symmetrizations of

Holder continuous functions.

ii

Acknowledgements

I would like to thank Professor Burchard for her wonderful supervision throughout the

completion of this thesis. On the professional side, I would like to thank her for intro-

ducing me to some interesting problems and letting me work independently while also

providing help whenever I needed it. On the personal side, her positive attitude and

good humour always made me feel comfortable around her. Lastly, I would like to thank

NSERC for their financial support over the last two years.

iii

Contents

1 Introduction 1

1.1 Steiner Symmetrization and the Isoperimetric Problem . . . . . . . . . . . . 1

1.2 Three Convergence Problems for Rearrangements . . . . . . . . . . . . . . . 2

2 Fundamental Properties of Rearrangements 12

2.1 Premilinaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Introduction to Rearrangements . . . . . . . . . . . . . . . . . . . . . 13

2.2 Rearrangement Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Compactness Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Explicit Universal Approximation 22

3.1 Sequences of Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Characterization of f∗ via Polarizations . . . . . . . . . . . . . . . . . 22

3.1.2 Metrics on Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.3 Uniform Convergence in Cc(Rd) and Convergence in Lp(Rd). . . . . 25

3.2 Sequences of Steiner Symmetrizations . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 Uniform Convergence in Cc(Rd) and Convergence in Lp(Rd). . . . . 29

3.3 Convergence in Hausdorff Distance . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Almost Sure Convergence 32

iv

4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1.1 Random Variables Distributed in Ω . . . . . . . . . . . . . . . . . . . 32

4.1.2 Random Variables Distributed in Sd−1 . . . . . . . . . . . . . . . . . . 33

4.2 Random Sequences of Polarizations . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Random Sequences of Steiner Symmetrizations . . . . . . . . . . . . . . . . . 38

4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Rates of Convergence: Holder Continuous Functions 46

5.1 Polarization of Holder Continuous Functions . . . . . . . . . . . . . . . . . . 46

5.2 Steiner Symmetrization of Holder Continuous Functions . . . . . . . . . . . 55

6 Rates of Convergence: Convex Bodies 57

6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.1.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.1.2 Notation and Preliminary Facts . . . . . . . . . . . . . . . . . . . . . . 58

6.1.3 Outline of the Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.2 Minkowski Symmetrizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.2.1 Notation and Preliminary Facts . . . . . . . . . . . . . . . . . . . . . . 61

6.2.2 Tools from Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . 63

6.2.3 L2 Decay of Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . 70

6.2.4 L∞ Decay of Spherical Harmonics . . . . . . . . . . . . . . . . . . . . 73

6.2.5 Final Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.3 Steiner and Minkowski Symmetrization: Making the Link. . . . . . . . . . . 79

6.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.3.2 Differential Geometry of Convex Bodies . . . . . . . . . . . . . . . . . 80

6.3.3 Inequalities between Quermassintegrals . . . . . . . . . . . . . . . . . 94

6.3.4 The Key Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.4 Steiner Symmetrizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

v

6.4.1 Circumradius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4.2 Inradius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.4.3 Final Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Bibliography 106

vi

Chapter 1

Introduction

1.1 Steiner Symmetrization and the Isoperimetric

Problem

Steiner symmetrization was invented by Jacob Steiner to solve the isoperimetric problem.

The isoperimetric problem in the plane says that among all closed and simple curves of

fixed length, the circle and only the circle encloses the most area. By using simple scaling

arguments, one can show that the isoperimetric problem is equivalent to showing that

among all sets of fixed area with simple closed curves as boundaries, the disc and only

the disc has minimum boundary length or minimum perimeter. Steiner first argued that

one could concentrate on convex compact sets (convex bodies) since the convex hull of

a non-convex set had larger area but smaller perimeter. He then showed that applying

a Steiner symmetrization to a convex body always strictly decreased perimeter while

preserving area unless that very same symmetrization did not change the initial convex

body. Since Steiner symmetrization preserves convexity and any convex body that is

invariant under all Steiner symmetrizations is necessarily a disc then Steiner concluded

that the isoperimetric problem was solved. His colleagues quickly objected to his proof

claiming that it was incomplete since he had failed to prove that a minimizer existed in

1

Chapter 1. Introduction 2

the first place. This is analogous to the claim that a smooth function is minimized at a

point simply because its derivative vanishes only at that point.

Today, we know that Steiner’s proof can be completed by appealing to a convergence

result for Steiner symmetrization namely that given any initial convex body in the plane

there exists a sequence of Steiner symmetrization that transforms it (in the Hausdorff

metric sense) to a disc of equal area. Since perimeter is continuous with respect to the

Hausdorff metric on the space of convex bodies, then any convex body of fixed area has

perimeter greater or equal to that of a disc of equal area. This shows that a perimeter

minimizing body exists and (by Steiner’s argument) is uniquely the disc.

The example above highlights one of the important applications of convergence re-

sults for rearrangements namely that the Schwartz symmetrization of convex bodies (in

any dimension) can be approximated by Steiner symmetrizations of convex bodies. The

main focus of this thesis is to give a thorough presentation of old and new convergence re-

sults for rearrangements. We focus on three rearrangements and the connection between

them: polarization, Schwartz symmetrization and Steiner symmetrization. A recurring

theme in this thesis is to approximate Schwartz and Steiner symmetrization by polariza-

tion (the simplest rearrangement). It is often the case that one can prove properties of

Schwartz and Steiner symmetrization, which seem at first difficult to prove, by approxi-

mating these rearrangements by polarizations and proving that the analogous properties

for polarization hold (see [3] for many applications of this approach). The content of

the thesis is motivated by three convergence problems for rearrangements that we now

describe in detail.

1.2 Three Convergence Problems for Rearrangements

In what follows, fσ will denote the polarization of f with respect to the reflection σ and

Su(f) will denote the Steiner symmetrization of f with respect to u. Furthermore, for


any finite sequence of unit vectors uini=1, we let

Su1,...,un(f) = (ni=1Sui) (f). (1.1)

For more on notation and general background on rearrangements see section 2.1 (“Pre-

liminaries”).

Problem 1: Explicit Universal Approximation.

Uniform Convergence for Cc(Rd)

Does there exist explicit sequences of reflections σi∞i=1 and unit vectors ui∞i=1 such that

limn→∞

∥fσ1⋯σn − f∗∥∞ = 0 (1.2)

and

limn→∞

∥Su1,...,un(f) − f∗∥∞ = 0 (1.3)

for every f ∈ Cc(Rd)? In Chapter 3 (“Explicit Universal Approximation”), we give a

self-contained exposition of a result of Van Schaftingen’s [16] that not only proves the

existence of sequences σi∞i=1 satisfying (1.2) but yields an algorithm for constructing

such sequences. The reader should note that in the following theorem, Ω denotes the

space of reflections across hyperplanes not containing the origin. In subsection 3.1.2

(“Metrics on Ω”), we construct a metric ρ on Ω.

Theorem 1. [17] Let σn∞n=1 ⊂ Ω be a dense subset of Ω with respect to the metric ρ. If

f ∈ Cc(Rd) and

fn+1 =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

fσ1 n = 0

fσ1⋯σn+1n n ≥ 1

then fn converges uniformly to f∗. In particular, the entire sequence fσ1σ2σ1σ2σ3⋯ con-


verges uniformly to f∗ for every f ∈ Cc(Rd) and in Lp(Rd) for every f ∈ Lp(Rd) with

1 ≤ p <∞.

In order to prove Theorem 1, Van Schaftingen uses certain properties of the polariza-

tion rearrangement. However, it can be shown (see Section 3.2) that analogous properties

also hold for Steiner symmetrization and consequently we extend Theorem 1 to Steiner

symmetrization:

Theorem 2. Let un∞n=1 ⊂ Sd−1 be a dense subset of Sd−1 with respect to the Euclidean

metric. If f ∈ Cc(Rd) and

fn+1 =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Su1(f) n = 0

Su1,...,un+1(fn) n ≥ 1

then fn converges uniformly to f∗. In particular, the entire sequence

Su1(f), Su1,u2(f), Su1,u2,u1,u2,u3(f), . . .

converges uniformly to f∗ for every f ∈ Cc(Rd) and in Lp(Rd) for every f ∈ Lp(Rd) with

1 ≤ p <∞.

Convergence in Hausdorff Distance

Let δ(⋅, ⋅) denote the Hausdorff distance. Does there exist explicit sequences of reflections

σi∞i=1 and unit vectors ui∞i=1 such that

limn→∞

δ(F σ1⋯,σn , F ∗) = 0 (1.4)

and

limn→∞

δ(Su1,...,un(F ), F ∗) = 0 (1.5)


for every compact set F ? In section 11 (“Convergence in Hausdorff Distance ”), it is

shown that the sequences satisfying properties 1.2 and 1.3 also satisfy properties 1.4 and

1.5. The essential idea is to (by using distance functions) express each compact set F

as a level set of a compactly supported continuous function. Then, by appealing to the

properties 1.2 and 1.3, one can show that 1.4 and 1.5 also holds.

Problem 2: Almost Sure Convergence.

The Basic Problem

In a paper of Mani-Levitska [11], it was shown that if Yi is a sequence of independent

random variables distributed uniformly on Sd−1 then with probability 1,

limn→∞

δ(SY1,...,Yn(K),K∗) = 0 (1.6)

for every compact convex set K. The very same author asked whether such a result

holds for arbitrary compact sets and this problem became known as the “Mani-Levitska

Conjecture”.

In Chapter 4 (“Almost Sure Convergence”), we consider the more general problem of

understanding which sequences of random variables Xi and Yi distributed in Ω and Sd−1

respectively satisfy the properties

P ( limn→∞

∥fX1⋯Xn − f∗∥∞ = 0 ∀f ∈ Cc(Rd)) = 1 (1.7)

and

P ( limn→∞

∥SY1,...,Yn(f) − f∗∥∞ = 0 ∀f ∈ CcRd)) = 1. (1.8)

By our previous discussion regarding convergence in Hausdorff distance, we know

that if 1.8 holds when Yi are independent and uniformly distributed in Sd−1 then the

Mani-Levitska conjecture also holds.


Existing Results

This Mani-Levitska conjecture appears to be settled first by Van Schaftingen [17]. We

briefly outline his result and the method of proof. Suppose that ρ1 and ρ2 are metrics

on Ω and Sd−1 respectively with the property that for every fixed f ∈ Cc(Rd), the map

f ↦ fσ and the map f ↦ Su(f) is continuous in Lp then the following holds:

Theorem 3. [17] If Xi and Yi are sequences of independent random variables distributed

in Ω and Sd−1 respectively satisfying the properties

lim infn→∞

P (Xn ∈ Bρ1(σ,λ)) > 0 (1.9)

and

lim infn→∞

P (Yn ∈ Bρ2(u,λ)) > 0 (1.10)

for all (σ,u) ∈ Ω × Sd−1 and for all λ > 0 then

P ( limn→∞

∥fX1⋯Xn − f∗∥∞ = 0 ∀f ∈ Cc(Rd)) = 1 (1.11)

and

P ( limn→∞

∥SY1,...,Yn(f) − f∗∥∞ = 0 ∀f ∈ CcRd)) = 1. (1.12)

To prove Theorem 3, Van Schaftingen first considered sequences σi∞i=1 ⊂ Ω and

ui∞i=1 ⊂ Sd−1 such that

limn→∞

∥fσ1⋯σn − f∗∥∞ = 0 (1.13)

and

limn→∞

∥Su1,...,un(f) − f∗∥∞ = 0 (1.14)

for every f ∈ Cc(Rd). Theorems 1 and 2 show that such sequences exist. He then showed


that (1.13) and (1.14) also holds for all sequences

τi∞i=1 ∶ ∀ε > 0,∀m ≥ 1,∃k ∋ ρ1(τk+i, σi) ≤ ε ∀1 ≤ i ≤m (1.15)

and

vi∞i=1 ∶ ∀ε > 0,∀m ≥ 1,∃k ∋ ρ2(vk+i, ui) ≤ ε ∀1 ≤ i ≤m (1.16)

and that these sequences had probability 1 under the conditions given in Theorem 3.

Volcic [18] also obtained a result about random sequences of Steiner symmetrizations.

In that paper, it is claimed that if Yi is a sequence of pairwise independent random

variables distributed uniformly in Sd−1 then

P ( limn→∞

∥SY1,...,Yn(f) − f∗∥p = 0 ∀f ∈ Lp(Rd)) = 1 (1.17)

for all 1 ≤ p < ∞. Volcic main probabilistic tool is the Borel-Cantelli Lemma which

holds for pairwise independent random variables (see [5, pp.50-51]). However, we are not

entirely convinced of the correctness of the use of the Borel-Cantelli Lemma in Volcic’s

paper. As a result, it is not clear whether pairwise independence is enough to obtain

1.17. That being said, the proof of the new results on almost sure convergence that is

presented in this thesis were partially inspired by Volcic’s paper.

New Results

In Chapter 4, we extend Theorem 3 by showing that the conditions on the random

variables Xi and Yi can be weakened. More precisely, we will show the following:

Theorem 4. If Xi∞i=1 is a sequence of independent random variables distributed in Ω

with the property that for every bounded sequence of points xi∞1 ⊂ Rd and for every

0 < λ < ∣xi∣ for all i ≥ 1∞∑i=1

µi(xi,Bi) =∞ (1.18)


for every sequence of balls Bi ⊂ (∣xi∣ − λ)Bd of uniform radius then

P ( limn→∞

∥fX1⋯Xn − f∗∥∞ = 0 ∀f ∈ Cc(Rd)) = 1 (1.19)

and

P ( limn→∞

∥fX1⋯Xn − f∗∥p = 0 ∀f ∈ Lp(Rd)) = 1. (1.20)

Theorem 5. If Yi is a sequence of independent random variables distributed in Sd−1 with

the property∞∑i=1

P (Yi ∈ B(ui, λ)) (1.21)

for every sequence ui∞i=1 ⊂ Sd−1 and for every λ > 0 then

P ( limn→∞

∥SY1,...,Yn(f) − f∗∥∞ = 0 ∀f ∈ Cc(Rd)) = 1 (1.22)

and

P ( limn→∞

∥SY1,...,Yn(f) − f∗∥p = 0 ∀f ∈ Lp(Rd)) = 1. (1.23)

In the last section of Chapter 4 (“Examples”) it is shown that if the conditions 1.9

and 1.10 of Theorem 3 are satisfied then so are the conditions 1.18 and 1.21 of Theorems

4 and 5 (see Propositions 12 and 13). Lastly, to show that Theorems 4 and 5 truly extend

Theorem 3, we give examples of sequences of independent random variables Xi and Yi

such that the conditions 1.18 and 1.21 of Theorems 4 and 5 are satisfied but not the

conditions 1.9 and 1.10 of Theorem 3.

Problem 3: Rates of Convergence.

Theorems 4 and 5 show that almost every random sequence of polarizations and Steiner

symmetrizations will transform (in the uniform convergence sense) any compactly sup-

ported continuous function into its corresponding decreasing symmetric rearrangement


and will also transform (in the Hausdorff distance sense) any compact set into its corre-

sponding Schwartz symmetrization. A natural question arises: how fast can convergence

occur? More precisely, let C ⊂ Cc(Rd) be a collection of functions that is invariant under

polarization or Steiner symmetrization i.e., fσ ∈ C for all (f, σ) ∈ C ×Ω or Su(f) ∈ C for

all (f, u) ∈ C ×Sd−1. Given ε > 0, what is the minimal number of polarizations (or Steiner

symmetrizations) needed to transform every function f ∈ C into a new function f ′ ∈ C

satisfying the property ∥f −f∗∥∞ ≤ ε ? Similarily, let A be a collection of compact sets in

Rd that is invariant under polarization or Steiner symmetrization. Given ε > 0, what is

the minimal number of polarizations (or Steiner symmetrizations) needed to transform

every compact set F ∈ A into a new compact set F ′ with the property δ(F ′, F ∗) ≤ ε?

Existing Results

To our knowledge, there are no published results on rates of convergence for polarizations.

There are, however, some very nice results regarding rates of convergence for Steiner

symmetrizations of convex bodies. The following result, due to Klartag [9], is currently

the best in the literature:

Theorem 6 (Theorem 1.5). [9]There exists a numerical constant C such that for every

0 < ε < 1, we need at most

⌈Cd4 log2(1/ε)⌉

Steiner symmetrizations to transform an arbitrary convex body K with volume κd into a

convex body K ′ with the property

(1 − ε)Bd ⊂K ′ ⊂ (1 + ε)Bd.

The final Chapter (“Rates of Convergence: Convex Bodies ”) is dedicated entirely to

giving a self-contained exposition of Theorem 6. Apart from providing all the necessary

background in spherical harmonics and differential geometry, many proofs are presented


differently and some are simplified. The most notable difference being the presentation

of the L∞ decay of spherical harmonics and related combinatorial estimates (see section

6.2.4 “L∞ Decay of Spherical Harmonics”).

New Results

In Chapter 5, we provide rates of convergence for both polarization and Steiner sym-

metrizations applied to particular subsets C of Cc(Rd). We denote by C0,α(A) the space

of all Holder continuous functions with exponent 0 < α ≤ 1 defined on a set A ⊂ Rd i.e.,

the space of all functions f with the property that there exists some positive constant c

such that

∣f(x) − f(y)∣ ≤ c∣x − y∣α. (1.24)

If f ∈ C0,α(A) then we let

[f]α = sup∣f(x) − f(y)∣∣x − y∣−α ∶ x ≠ y. (1.25)

In our applications, the domain of f will always be fixed and so there need not be any

domain dependence for the symbol [f]α. The following is proved:

Theorem 7. There exists an explicit constant C(α, d, λ1, λ, λ2) such that for all ε > 0

there exists at most

⌈C(α, d, λ1, λ, λ2)(1/ε)(1+(d+1)/α)(1+ dα)⌉ (1.26)

polarizations that transform any function f ∈ C0,α(λBd) with [f]α ≤ λ1 and ∥f−f∗∥∞ ≤ λ2

into f ′ with the property

∥f ′ − f∗∥∞ ≤ ε. (1.27)

Theorem 8. For all ε > 0 there exists at most

⌈C(α, d, λ1, λ, λ2)(1/ε)(1+(d+1)/α)(1+ dα)⌉ (1.28)


Steiner symmetrizations that transform any function f ∈ C0,α(λBd) with [f]α ≤ λ1 and

∥f − f∗∥∞ ≤ λ2 into f ′ with the property

∥f ′ − f∗∥∞ ≤ ε. (1.29)

The constant C(α, d, λ1, λ, λ2) is the same in both Theorems and will be computed

explicitly in the proof of Theorem 7.

Unlike Theorem 6, the rates of convergence in Theorems 7 and 8 are not exponential.

They do, however, constitute a first step towards understanding rates of convergence for

rearrangements of particular subsets of Cc(Rd). Lastly, we believe that Theorems 7 and 8

combined with the method of proof of Proposition 11, can be used to obtain rates of con-

vergence for polarizations and Steiner symmetrizations of compact sets. Unfortunately,

we did not have time to investigate this further.

Chapter 2

Fundamental Properties of

Rearrangements

2.1 Premilinaries

2.1.1 Notation

Given any metric ρ, Bρ(x,λ) will denote the closed ball of radius λ centered at x i.e., the

set of all points whose distance from x (with respect to the metric ρ) is at most λ. For

convenience, when ρ is the Euclidean metric we will denote a ball of radius λ centered at

x by B(x,λ). In addition, Bd and Sd−1 will denote the unit ball and unit sphere in Rd.

The volume and surface area of Bd will be denoted by κd and γd respectively.

Given any two real numbers a and b, a ∧ b and a ∨ b will denote the minimum and

maximum of a and b respectively. If a is any real number then a+ = a∨ 0 and a− = −a∨ 0

will denote the positive and negative part of a respectively.

The space of continuous functions of compact support will be denoted by Cc(Rd). If

f ∈ Cc(Rd), we let

ω(f, h) = sup∣f(x) − f(y)∣ ∶ ∣x − y∣ ≤ h (2.1)

12

Chapter 2. Fundamental Properties of Rearrangements 13

denote the modulus of continuity of f an define the “inverse”of ω(f, h) by

ω−1(f, t) = suph ∶ ω(f, h) ≤ t. (2.2)

M will denote the sigma-algebra consisting of all Lebesgue measurable subsets of Rd

and m will denote the corresponding Lebesgue measure. We suppress the dependence of

the symbols M and m on the dimension d since it will always be clear from the context

which dimension d applies. We also let θ denote the normalized surface area measure on

Sd−1.

Given any set A, χ(A) will denote its characteristic function:

χ(A) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1 x ∈ A

0 x ∉ A(2.3)

. We will often encounter set mappings Ξ on level sets x ∶ f(x) > t for some t and f .

Instead of writing Ξ(x ∶ f(x) > t), we will simply write Ξ(f > t). As an example, if µ is

a measure for which x ∶ f(x) > t is measurable for some t and f then we write µ(f > t)

to denote µ(x ∶ f(x) > t).

We define Ω to be the collection of all reflections through hyperplanes not containing

0 in Rd. Arbitrary elements of Ω are denoted by σ. Lastly, if σ is a reflection through a

hyperplane Xσ0 then this hyperplane splits Rd into two open half-spaces. The half-space

containing zero will be denoted by Xσ+ and the other by Xσ

− .

2.1.2 Introduction to Rearrangements

We now introduce the concept of rearrangements of sets and functions. Our presentation

follows that of [3]. A rearrangement T is a map T ∶ M → M that is both monotone

(A ⊂ B implies T (A) ⊂ T (B)) and measure preserving (m(T (A)) =m(A) for all A ∈M).


We say that a non-negative measurable function f vanishes at infinity if

m(f > t) <∞ (2.4)

for all t > 0. If f vanishes at infinity then we can define its rearrangement Tf by using

the “layer cake principle”

Tf(x) = ∫∞

0χT (f>t)(x)dt = supt ∶ x ∈ T (f > t). (2.5)

We immediately see that

∞⋃i=1

T (f > t + 1/n) ⊂ Tf > t ⊂ T (f > t) (2.6)

for all t ≥ 0. However, we also have

m(∞⋃i=1

T (f > t + 1/n)) = limn→∞

m(T (f > t + 1/n))

= limn→∞

m(f > t + 1/n)

= m(f > t)

for all t ≥ 0 and thus

m(f > t) =m(Tf > t) (2.7)

for all t ≥ 0.

In this thesis, we will be primarily concerned with three types of rearrangements:

polarization, symmetric decreasing rearrangement and Steiner symmetrization.


Polarization

Let σ ∈ Ω and let f be an arbitrary function. We define the polarization of f with respect

to σ as

fσ(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

f(x) ∨ f(σ(x)) if x ∈Xσ+

f(x) ∧ f(σ(x)) if x ∈Xσ−

f(x) if x ∈Xσ0

(2.8)

. If A is an arbitrary set then its polarization with respect to σ is simply the polarization

of χA and is denoted by Aσ:

Aσ = (σ(A ∩Xσ−) ∩Ac) ∪ (σ(A ∩Xσ

+) ∩A) ∪ (A ∩Xσ+) ∪ (A ∩Xσ

0 ) . (2.9)

In other words, Aσ is the same as A except that the part of A contained in Xσ− whose

reflection does not lie in A is replaced by its reflection in Xσ+ . As a result, we see

that polarization is measure preserving. It is clear from 2.8 that if f ≤ g then fσ ≤ gσ

for all σ ∈ Ω and in particular polarization is monotone on M i.e., polarization is a

rearrangement. One can check directly that fσ > t = f > tσ for all σ ∈ Ω and in

particular (recalling 2.5) fσ is the rearrangement of f with respect to the polarization

rearrangement.

Symmetric Decreasing Rearrangement

For any A ∈M there exists a unique open ball centered at the origin A∗ with the same

measure as A. A∗ is called the Schwartz symmetrization of A. If f(x) vanishes at infinity

then its rearrangement with respect to the Schwartz rearrangement is denoted by f∗(x).

It is clear that f∗(x) is radially decreasing i.e., f∗(x) ≤ f∗(y) if ∣x∣ ≥ ∣y∣ and f(x) = f(y)

if ∣x∣ = ∣y∣. In the literature, f∗ is also called the symmetric decreasing rearrangement of


f. If f vanishes at infinity, we let

rf(t) = (m(f > t)κd

)1/d

(2.10)

denote the radius of the open ball f > t∗. The distribution function of a function f

vanishing at infinity is always right continuous and thus so is rf(t). In particular, we

have

f∗ > t = f > t∗ (2.11)

for all t ≥ 0 and thus f∗ is right continuous.

Steiner Symmetrization

If u ∈ Sd−1 and A ∈M then, by Fubini’s theorem, the set

Ax′ = ⟨x,u⟩ ∶ x′ + u ⟨x,u⟩ ∈ A (2.12)

is measurable in R for almost every x′ ∈ u⊥. If we let A∗x′ equal the empty set whenever

Ax′ is non-measurable then we denote by

Su(A) = ⋃x′∈u⊥

x′ + uA∗x′ (2.13)

the Steiner symmetrization of A with respect to u. It is clear that Steiner symmetrization

is monotone and, by Fubini’s theorem, measure preserving. If f vanishes at infinity we

denote by Su(f) the rearrangement of f with respect to the rearrangement Su or simply

the Steiner symmetrization of f with respect to u. Lastly, we let

Su1,...,un(f) = (ni=1Sui) (f) (2.14)

for any finite sequence of unit vectors uini=1.


Now that we have introduced rearrangements, we warn the reader that whenever

we speak of a rearrangement of f then one should always assume that f is both van-

ishing at infinity and non-negative. There will thus be no need to continuously state

that the function must be vanishing at infinity and non-negative prior to considering its

rearrangement.

2.2 Rearrangement Inequalities

Proposition 1. (Polarization Formula for Functionals)If σ ∈ Ω then

∫Rd

[fσgσ − fg]dx = ∫Xσ±

[(f(x) − f(σ(x))) (g(x) − g(σ(x)))]−dx

for any two functions f and g.

Proof. If a1, a2, b1, b2 are real numbers with a1 ≤ a2 and b1 ≤ b2 then

(a1b1 + a2b2) − (a1b2 + a2b1) = (a1 − a2)(b1 − b2) ≥ 0. (2.15)

Given x ∈ Rd, we let

a1(x) = f(x) ∧ f(σ(x)) , a2(x) = f(x) ∨ f(σ(x))

b1(x) = g(x) ∧ g(σ(x)) , b2(x) = g(x) ∨ g(σ(x)).

Recalling the definition of polarization of functions (see 2.8), one has

fσ(x)gσ(x) + fσ(x)gσ(σ(x)) = a1(x)b1(x) + a2(x)b2(x). (2.16)

If (f(x) − f(σ(x))) (g(x) − g(σ(x))) < 0 then we must have

(f(x) − f(σ(x))) (g(σ(x)) − g(x)) = (a1(x) − a2(x))(b1(x) − b2(x)) > 0. (2.17)


It is clear that (f(x) − f(σ(x))) (g(x) − g(σ(x))) ≥ 0 if and only if (fσgσ − fg)(x) +

(fσgσ − fg)(σ(x)) = 0. By 2.15,2.16 and 2.17, we must have

(fσgσ − fg)(x) + (fσgσ − fg)(σ(x)) = [(f(x) − f(σ(x))) (g(x) − g(σ(x)))]−. (2.18)

Integrating both sides of 2.18 over either Xσ+ or Xσ

− and using the fact that σ is an

isometry yields

∫Rd

[(fσgσ)(x) − (fg)(x)]dx = ∫Xσ±

[(f(x) − f(σ(x))) (g(x) − g(σ(x)))]−dx ≥ 0.

Corollary 1. Let f ∈ Cc(Rd) and let g be any integrable and strictly radially decreasing

function i.e., g(x) > g(y) if ∣x∣ < ∣y∣. Then for every σ ∈ Ω, fσ = f if and only if

∫Rdf(x)g(x)dx = ∫

Rdfσ(x)g(x)dx. (2.19)

Proof. If 2.19 holds then by Proposition 1, we have f(x) ≥ f(σ(x)) for all x ∈Xσ+ which

is the same as fσ = f .

Proposition 2. (Hardy-Littlewood Inequality) If T is a rearrangement then

∫Rdfgdx ≤ ∫

RdT (f)T (g)dx. (2.20)

for any two functions f and g.

Proof. We first note that any non-negative function h has the following “layer cake”

representation

h(x) = ∫∞

0χh>t(x)dt. (2.21)


In particular, we have by 2.21 and Fubini’s theorem


∞

0∫

∞

0m(f > t ∩ g > s)dtds. (2.22)

It thus suffices to prove that for every pair of non-negative real numbers s and t

m(f > t ∩ g > s) ≤m(T (f) > t ∩ T (g) > s). (2.23)

By the measure preserving and monotonicity properties of T , we have

m(f > t ∩ g > s) =m(T (f > t ∩ g > s)) ≤m(T (f > t) ∩ T (g > s)). (2.24)

However, recalling 2.6 and 2.7, we must have

m(T (f > t) ∩ T (g > s)) =m(Tf > t ∩ Tg > s). (2.25)

Using 2.24 and 2.25 yields 2.23 and completes the proof.

Proposition 3. (Lp Contraction) If 1 ≤ p <∞ and T is any rearrangement then

∥T (f) − T (g)∥p ≤ ∥f − g∥p

for any (f, g) ∈ Lp(Rd) ×Lp(Rd).

Proof. The idea will be to follow the same line of reasoning that was used to prove the

Hardy-Littlewood inequality i.e., we want to show that the validity of the Proposition

follows from the inequality 2.23. We first observe that ∣f(x) − g(x)∣p equals

p∫∞

0[(f(x) − t)p−1

χg≤tχf>t(x) + (g(x) − t)p−1χf≤tχg>t(x)]dt. (2.26)


We have χg≤t(x) = 1 − χg>t(x) and similarily for f . In addition, we have

∫Rd∫

∞

0p(f(x) − t)p−1χf>t(x)dtdx = ∥f∥pp = ∥T (f)∥pp (2.27)

and similarily for g. By considering the layer cake representation of (f(x) − t)p−1 and

that of (g(x) − t)p−1 and by applying Fubini’s theorem, it thus suffices to prove that for

any positive real numbers s, t

m ((f − t)p−1 > s ∩ g > t) ≤m ((T (f) − t)p−1 > s ∩ T (g) > t) (2.28)

and

m ((g − t)p−1 > s ∩ f > t) ≤m ((T (g) − t)p−1 > s ∩ T (f) > t) . (2.29)

However, 2.28 and 2.29 easily follow from 2.23.

By taking p→∞ in the last Proposition, we obtain:

Corollary 2. (L∞ Contraction) If T is any rearrangement then

∥T (f) − T (g)∥∞ ≤ ∥f − g∥∞

for all (f, g) ∈ Cc(Rd) ×Cc(Rd).

2.3 Compactness Properties

Proposition 4. Let f ∈ Cc(Rd) with modulus of continuity ω(f, h) then ω(fσ, h) ≤

ω(f, h) for all σ ∈ Ω.

Proof. Let x and y be two points in Rd with ∣x − y∣ ≤ h. It is clear that ∣fσ(x) − fσ(y)∣

equals one of the following four values :

∣f(x) − f(y)∣, ∣f(σ(x)) − f(σ(y))∣, ∣f(σ(x)) − f(y)∣, ∣f(x) − f(σ(y))∣. (2.30)


Since σ is an isometry, it suffices to consider only the last two cases in 2.30. If x, y

belong to different half spaces Xσ± then ∣x − σ(y)∣ = ∣σ(x) − y∣ < ∣x − y∣ and so the last

two cases of 2.30 are bounded by ω(f, h). We are thus left with the case where x, y

belong to the same half space Xσ± and the last two cases of 2.30 hold. In general for any

four real numbers a1, a2, b1, b2 with a1 ≤ a2 and b1 ≤ b2 we have ∣a2 − b2∣ ≤ ∣a1 − b2∣ and

∣a1 − b1∣ ≤ ∣a1 − b2∣. In particular ∣f(x) − f(σ(y))∣ and ∣f(σ(x)) − f(y)∣ are bounded by

∣f(x) − f(y)∣ ≤ ω(f, h).

Corollary 3. Let σn∞n=1 ⊂ Ω then fσ1⋯σn∞n=1 is precompact in Cc(Rd) (with respect to

uniform convergence) for every f ∈ Cc(Rd).

Proof. It is clear that each element of fσ1⋯σn∞n=1 is bounded by ∥f∥∞. If f has support

in λBd for some λ > 0 then

fσ1⋯σn > 0 = f > 0σ1⋯σn ⊂ (λBd)σ1⋯σn = λBd.

By the previous Proposition, fσ1⋯σn∞n=1 is an equicontinuous family and thus, by the

Arzela-Ascoli theorem, the sequence fσ1⋯σn∞n=1 has a subsequence which converges uni-

formly to a function f in Cc(Rd).

Chapter 3

Explicit Universal Approximation

3.1 Sequences of Polarizations

3.1.1 Characterization of f∗ via Polarizations

Proposition 5. Let f ∈ Cc(Rd) then f∗ is also in Cc(Rd). Furtheremore, f = f∗ if and

only if fσ = f for all σ ∈ Ω.

Proof. We first show that if f ∈ Cc(Rd) then f∗ ∈ Cc(Rd). If f∗(r) denotes the value of f∗

at ∣x∣ = r then because ∣∣x∣−∣y∣∣ ≤ ∣x−y∣, it suffices to show that f∗(r) is continuous. In fact,

since f∗(r) is always right continuous it suffices to prove that f∗(r) is left continuous. If

this is false then there is a jump at a point r1 > 0 i.e., there exists ε > 0 such that for all

r < r1, f∗(r) − f∗(r1) ≥ ε. In particular, we have

m(f∗(r1) < f(x) < f∗(r1) + ε) = ∅ (3.1)

and this would contradict the continuity of f so f∗ is also continuous.

To prove the second part of the Proposition, we first observe that f is radially de-

creasing if and only if

(f(x) − f(σ(x))(∣x∣ − ∣σ(x)∣) ≥ 0 (3.2)

22

Chapter 3. Explicit Universal Approximation 23

for all x and for all σ ∈ Ω. However, 3.2 is also equivalent to fσ = f for all σ ∈ Ω and thus

fσ = f for all σ ∈ Ω is equivalent to f being radially decreasing. It thus suffices to show

that if f is radially decreasing then f = f∗. If this is the case then f > t∗ = f > t

since f > t is an open ball and this in turn yields

f∗(x) = supt≥0

x ∈ f > t∗ = supt≥0

x ∈ f > t = f(x) (3.3)

for all x.

3.1.2 Metrics on Ω

Given any two points x and y with ∣x∣ ≠ ∣y∣, we let σx,y denote the unique reflection that

maps x to y. In particular, the map

x↦ σ0,x (3.4)

establishes a one to one correspondence between Rd − 0 and Ω. In fact, one has

σ0,x(y) = y + ⟨x − 2y,x

∣x∣⟩ x

∣x∣(3.5)

As a consequence, if ρ′ is any metric on Rd then letting ρ(σ1, σ2) = ρ′(σ1(0), σ2(0)) yields

a metric ρ on Ω. The following Proposition shows that the metric

ρ(σ1, σ2) = ∣σ1(0) − σ2(0)∣ (3.6)

on Ω induces a pleasant continuity property.

Proposition 6. For every fixed f ∈ Cc(Rd) and for every 1 ≤ p <∞, the map f ↦ fσ is

continuous with respect to the metric ρ(σ1, σ2) = ∣σ1(0) − σ2(0)∣ on Ω and the Lp norm

on Cc(Rd).


Proof. Let σ1, σ2 ∈ Ω and suppose that the support of f is contained in λBd for some

λ > 0. From 3.5, we deduce that ∣σ2(x) − σ1(x)∣ is bounded by

∣σ1(0) − σ2(0)∣ + 4λ ∣ σ1(0)∣σ1(0)∣

− σ2(0)∣σ2(0)∣

∣ (3.7)

for all x ∈ λBd. If we let

A(σ1, σ2, λ) = (Xσ1+ ∩Xσ2

− ) ∪ (Xσ1− ∩Xσ2

+ ) ∩ λBd

then because

∣(f(x) ∧ f(σ(x))) − (f(x) ∧ f(σ2(x)))∣ ≤ ∣f(σ1(x) − f(σ2(x))∣

and

∣(f(x) ∨ f(σ(x))) − (f(x) ∨ f(σ2(x)))∣ ≤ ∣f(σ1(x)) − f(σ2(x))∣

we obtain the inequality

∥fσ1 − fσ2∥pp ≤ 2∥f∥p∞m(A(σ1, σ2, λ)) + ∫λBd

∣f(σ1(x) − f(σ2(x))∣pdx. (3.8)

We note that A(σ1, σ2, λ) also equals

x ∶ ∣σ1(x)∣ < ∣x∣ < ∣σ2(x)∣, ∣x∣ ≤ λ ∪ x ∶ ∣σ2(x)∣ < ∣x∣ < ∣σ1(x)∣, ∣x∣ ≤ λ. (3.9)

By 3.7,3.9 and the uniform continuity of f on λBd, it is clear that the right-hand side of

the inequality 3.8 converges to zero as σ2 approaches σ1.


3.1.3 Uniform Convergence in Cc(Rd) and Convergence in Lp(Rd).

proof of Theorem 1. By Corollary 3, the sequence of functions fn is precompact in Cc(Rd)

and consequently we can find a subsequence σnk∞k=1 such that fnk converges uniformly

to h. We wish to show that h = f∗. We let

I(f) = ∫Rde−∣x∣f(x)dx. (3.10)

Let l ≥ 1 and choose k large enough that nk ≥ l. We have by the contraction property of

rearrangements (Proposition 3)

limk→∞

∥fσ1⋯σlnk− hσ1⋯σl∥∞ ≤ lim

k→∞∥fnk − h∥∞ = 0. (3.11)

By Proposition 1, the sequence I(fn) is increasing and thus by 3.11

I(hσ1⋯σl) = limk→∞

I(fσ1⋯σlnk) ≤ lim

k→∞I(fnk+1) = I(h). (3.12)

By Propositon 1, this implies hσl = h for all l ≥ 1 and by the density of the sequence

σl∞l=1 combined with Proposition 6, we obtain hσ = h for all σ ∈ Ω. By Proposition 5, we

conclude that h = h∗ and since h was an arbitrary limit point we have that fn converges

uniformly to h = h∗. The equimeasurability of the functions fn is not enough to conclude

that h∗ = f∗. The key is to use the contraction property (Corollary 2) for the symmetric

decreasing rearrangement. More precisely, we have

∥f∗ − h∥∞ ≤ ∥f∗ − f∗n∥∞ + ∥f∗n − h∗∥∞ = ∥f∗n − h∗∥∞ ≤ ∥fn − h∥∞ (3.13)

for all n ≥ 1 and by taking limits in 3.13, we obtain the desired result. The fact that

fσ1σ2σ1σ2σ3⋯ converges uniformly to f∗ follows directly from Corollary 2. If f ∈ Lp(Rd)

then we can find a sequence of functions fk ∈ Cc(Rd) such that fk converges in Lp(Rd)


to f . If τn∞n=1 denotes the sequence σ1, σ2, σ1, σ2, σ3, . . . then for all (k,n) ∈ N ×N

∥f τ1⋯τn − f∗∥p ≤ ∥f τ1⋯τn − f τ1⋯τnk ∥p + ∥f τ1⋯τnk − f∗k ∥p + ∥f∗k − f∗∥p.

≤ ∥f τ1⋯τnk − f∗k ∥p + 2∥fk − f∥p.

Sending n to infinity and then k to infinity yields the desired result.

Corollary 4. For any f ∈ Cc(Rd), ω(f∗, h) ≤ ω(f, h).

Proof. By Theorem 1, there exists a sequence σn∞n=1 such that fσ1⋯σn converges to f∗

uniformly for all f ∈ Cc(Rd). By Proposition 4, we have

ω(f∗, h) = limn→∞

ω(fσ1⋯σn , h) ≤ ω(f, h).

If m(E) <∞ then

PerM(E) = limε↓0

m(E + εBd) −m(E)ε

(3.14)

denotes the Minkowski perimeter of E whenever the limit 3.14 exists.

We will now use Corollary 4 to give a very short proof of the isoperimetric inequality.

Proposition 7. (Isoperimetric Inequality) For any measurable set E with finite Minkowski

perimeter and volume κd

PerM(E) ≥ γd. (3.15)

Proof. Consider the function f(x) = 1 − d(E,x) with d(E,x) the distance function from

E restricted to E +Bd. It is well known that f(x) is a contraction and clearly f(x) = 1

if and only if x ∈ E. By corollary 4, f∗ is also a contraction and in particular

lim inft↓0

r(1 − t) − 1

t≥ 1. (3.16)


However, we also have

limt↓0

r(1 − t) − 1

t= κ−1/d

d limt↓0

m(f > 1 − t)1/d − κ1/dd

t= γ−1

d PerM(E). (3.17)

Combining 3.16 and 3.17 completes the proof.

3.2 Sequences of Steiner Symmetrizations

Proposition 8. Let rn∞n=1 be any dense subset of R and form the sequence σn =

σ0,rnu∞n=1 ⊂ Ω. Then for all f ∈ Cc(Rd), fσ1σ2σ1σ2σ3⋯ converges uniformly to Su(f) and

ω(Su(f), h) ≤ ω(f, h) for all h > 0.

Proof. For any x′ ∈ u⊥, we let fx′(r) = f(x′ + ru). By Theorem 1 (applied to d = 1),

fσ1σ2σ1σ2σ3⋯(x′ + ru) converges pointwise to f∗x′(r) for every x′ ∈ u⊥. If x = x′ + x′′

with x′′ = ⟨x,u⟩u and we let g(x) = f∗x′(x′′) then by Corollary 3, fσ1σ2σ1σ2σ3⋯ converges

uniformly to g(x). Hence it suffices to show that g(x) = Su(f). Let t ≥ 0 then

g(x) > t = x ∶ f∗x′(x′′) > t = x′ + fx′(r) > t∗u = Su(f > t)

and thus

g(x) = supt≥0

x ∈ g > t = supt≥0

x ∈ Su(f > t) = Su(f)(x). (3.18)

The fact that ω(Su(f), h) ≤ ω(f, h) for all h > 0 follows from Proposition 4 and uniform

convergence of fσ1σ2σ1σ2σ3⋯ to Su(f).

Characterization of f∗ via Steiner Symmetrizations

Proposition 9. For any f ∈ Cc(Rd), f = f∗ if and only if Su(f) = f for all u ∈ Sd−1.


Proof. If f = f∗ then for all t ≥ 0 and for all u ∈ Sd−1

Su(f > t) = Su(f > t∗) = f > t∗ = f > t

and this implies (see 3.18) Su(f) = f . If Su(f) = f for all u ∈ Sd−1 then for all t ≥ 0

Su(f > t) = Su(f) > t = f > t, ∀u ∈ Sd−1.

We will show that if x1, x2 ∈ ∂f > t then ∣x1∣ = ∣x2∣. Let ui = xi/∣xi∣, x3 = 12(u1 + u2)

and let πx3 be the reflection across x⊥3 . Since Su3(f > t) = f > t then necessarily

πx3(x1) = x2 which in turn implies that ∣x1∣ = ∣x2∣. In other words, f > t is an open ball

and this implies (see 3.3) that f = f∗.

Corollary 5. If g is any integrable and strictly radially decreasing function and f ∈

Cc(Rd)then for every u ∈ Sd−1, Su(f) = f if and only if


RdSu(f)(x)g(x)dx. (3.19)

Furthermore, f = f∗ if and only if 3.19 holds for all u ∈ Sd−1.

Proof. By Proposition 8, there exists a sequence σn∞n=1 ∈ Ω∞ such that fσ1⋯σn converges

to Su(f) uniformly. If 3.19 holds then by Corollary 1, we must have fσn = f for all n ≥ 1

which in turn implies that Su(f) = f . If 3.19 holds for all u ∈ Sd−1 then by the previous

sentence we must have Su(f) = f for all u ∈ Sd−1 and by Lemma 9 we deduce that

f = f∗.

Proposition 10. For every fixed f ∈ Cc(Rd) and for every 1 ≤ p <∞, the map

u↦ Su(f) (3.20)

is continuous with respect to the Euclidean norm on Sd−1 and the Lp(Rd) norm on Cc(Rd).


Proof. If u1 ∈ Sd−1 then there exists rotations Qu such that Qu(u1) = u for all u ∈ Sd−1

and Qu(x) converges uniformly to x on compacts whenever u approaches u1. It is clear

that

Su(f) = Su1(f Q−1u ) Qu (3.21)

for all u ∈ Sd−1. In particular, ∥Su(f) − Su1(f)∥p is bounded by

∥Su1(f Q−1u ) Qu − Su1(f) Qu∥p + ∥Su1(f) Qu − Su1(f)∥p. (3.22)

By the contraction property of rearrangements (Proposition 3):

∥Su1(f Q−1u ) Qu − Su1(f) Qu∥p ≤ ∥f Q−1

u − f∥p. (3.23)

Combining 3.22 and 3.23, we have

∥Su(f) − Su1(f)∥p ≤ ∥f Q−1u − f∥p + ∥Su1(f) Qu − Su1(f)∥p

= ∥f − f Qu∥p + ∥Su1(f) Qu − Su1(f)∥p. (3.24)

As u approaches u1, fQu and Su1(f)Qu converge uniformly to f and Su1(f) respectively

and thus the right-hand side of 3.24 approaches zero.

3.2.1 Uniform Convergence in Cc(Rd) and Convergence in Lp(Rd).

proof of Theorem 2. If we examine the proof of Theorem 1, all that was needed were

the properties of polarization described in Corollaries 1 and 3 as well as those found in

Propositions 5 and 6. However, these properties are analogous to the properties found in

Proposition 8, Corollary 5 and Proposition 10 for Steiner symmetrization. As a result, it

is clear that the proof of Theorem 1 also works for Steiner symmetrization and we thus

obtain Theorem 2.


3.3 Convergence in Hausdorff Distance

Theorems 1 and 2 show that there exists a sequence of reflections σn∞n=1 and a sequence

of unit vectors un∞n=1 such that Aσ1...σn and Su1,...,un(A) converges in symmetric differ-

ence to A∗ for any A ∈M with m(A) < ∞. When F is a compact set, one can obtain

a stronger notion of convergence of the sets F σ1⋯σn and Su1,...,un(F ) to F ∗ by replacing

convergence in symmetric difference with convergence in the Hausdorff distance. We will

let δ define the Hausdorff distance on the collection of all subsets contained in Rd i.e.,

δ(A1,A2) = infε ∶ A1 ⊂ A2 + εBd,A2 ⊂ A1 + εBd (3.25)

for any two sets A1,A2.

Proposition 11. If Ti∞i=1 is a sequence of rearrangements and Tn ⋯T1(f) converges

uniformly to f∗ for every f ∈ Cc(Rd) then Tn⋯T1(F ) converges to F ∗ in the Hausdorff

distance for any compact set F .

Proof. As in the proof of Proposition 7, given any compact set F we let f(x) = 1−d(F,x)

with d(F,x) the distance function from F restricted to the set F + Bd. Since f is

contractive, we have fn ∶= Tn ⋯ T1(f) converging uniformly to f∗. Clearly fn(x) = 1

if and only if x ∈ Fn ∶= Tn ⋯ T1(F ). Choose xn in the closure of F ∗ such that

δ(F ∗, Fn) ≥ d(xn, Fn) with d(xn, Fn) denoting the distance between xn and Fn. Noting

that

m(F ∗ − Fn) =m(Fn − F ∗) (3.26)

yields

m (B (xn, δ(F ∗, Fn)) ∩ F ∗) ≤m(Fn −Bd). (3.27)

If ε > 0 and ∥fn − f∗∥∞ < ε then necessarily

Fn ⊂ x ∶ f∗(x) > 1 − ε (3.28)


and by 3.27:

m (B (xn, δ(F ∗, Fn)) ∩ F ∗) ≤m(f > 1 − ε) −m(F ) =m(f > 1 − ε) −m(f = 1). (3.29)

Since fn converges uniformly to f∗ then by 3.29, Fn converges to F ∗ in Hausdorff distance.

Chapter 4

Almost Sure Convergence

4.1 Preliminaries

4.1.1 Random Variables Distributed in Ω

Recalling subsection 3.1.2, we introduced a metric ρ on Ω by letting ρ(σ1, σ2) = ∣σ1(0) −

σ2(0)∣. Probability measures will always be defined on the Borel sigma-algebra generated

by the metric ρ. We will denote this sigma-algebra by F . It is clear that F contains

all sets of the form σ ∶ σ(x) ∈ U with U an open subset of Rd and x ∈ Rd. Given any

sequence of probability measures µi defined on (Ω,F), there exists, by the Kolmogorov

extension theorem [5, pp.471-473] , a unique probability measure P defined on the infinite

product space (ΩN,FN) such that

P (σ1 ∈ A1, . . . , σn ∈ An) =n

∏i=1

µi(Ai). (4.1)

We will denote in bold σσσ = (σ1, . . . , σn, . . .) arbitrary elements of ΩN. If we let Xi(σσσ) = σi

then by 4.1, Xi∞i=1 is a sequence of independent random variables distributed in Ω with

P (Xi ∈ A) = µi(A) for all i ≥ 1. Conversely, given any sequence of independent random

variables Xi distributed in Ω and defined on some fixed probability space (S,F1, P1), we

32

Chapter 4. Almost Sure Convergence 33

can construct a probability space (ΩN,FN, P ) by letting

P (σ1 ∈ A1, . . . , σn ∈ An) =n

∏i=1

µi(Ai) (4.2)

for all n ≥ 1 with µi(A) = P1(Xi ∈ A). We will thus view any sequence of independent

random variables Xi distributed in Ω as coordinate maps σi defined on probability spaces

of the form (ΩN,FN, P ) with P =⊗∞i=1 µi and we will let µi(x,A) denote the probability

that σi maps x into A i.e.,

µi(x,A) = µi(σi(x) ∈ A) = P (σi(x) ∈ A). (4.3)

4.1.2 Random Variables Distributed in Sd−1

As for sequences of independent random variables distributed in Ω, we will view any

sequence Yi of independent random variables distributed in Sd−1 as coordinate maps

defined on probability spaces of the form ((Sd−1)N,GN, P ) with G the Borel sigma-algebra

generated by the Euclidean metric restricted to Sd−1 and P =⊗∞i=1 µi for some sequence

of probability measures µi defined on (Sd−1,G). In other words, if uuu = (u1, . . . , un, . . .)

denotes arbitrary elements of the infinite product space (Sd−1)N then Xi(uuu) = ui and

P (Xi ∈ A) = µi(A) (4.4)

for every A ∈ G.

4.2 Random Sequences of Polarizations

Lemma 1. If f ∈ Cc(Rd) and ε(x) = ∣f(x) − f∗(x)∣ > 0 then there exists y ∈ Rd such that

∣∣w∣ − ∣z∣∣ ≥ ω−1(f∗, ε(x)/4) (4.5)


and

(∣w∣ − ∣z∣)(f(w) − f(z)) ≥ ω−1(f∗, ε(x)/4)ε(x)/4 (4.6)

for every

(w, z) ∈ B(x,ω−1(f, ε(x)/8)) ×B (y,ω−1(f, ε(x)/8)) .

Proof. If t ≥ f∗(x) then m(f(y) ≤ t, ∣y∣ < ∣x∣) equals

m(f∗(y) ≤ t) −m(f(y) ≤ t, ∣y∣ ≥ ∣x∣) (4.7)

which is bounded below by

m (f∗(y) ≤ t) −m(∣y∣ > ∣x∣) =m(f∗(y) ≤ t, ∣y∣ ≤ ∣x∣). (4.8)

Similarily, if 0 ≤ t < f∗(x) then m(f(y) > t, ∣y∣ ≥ ∣x∣) equals

m(f∗(y) > t) −m(f(y) > t, ∣y∣ < ∣x∣) (4.9)

which is bounded below by

m (f∗(y) > t) −m(∣x∣ > ∣y∣) =m(f∗(y) > t, ∣x∣ ≤ ∣y∣). (4.10)

If f∗(x) < t < f(x) then by the above

m(∣y∣ ≤ ∣x∣, f(y) ≤ t + s) ≥m(r(t + s) ≤ ∣y∣ ≤ ∣x∣) >m(r(t) ≤ ∣y∣ ≤ ∣x∣) (4.11)

for every t + s ≤ f(x). This implies that for every such s, there exists y(s) such that

∣y(s)∣ ≤ r(t) and f(y(s)) ≤ t + s. By continuity of f , we deduce that there exists y(t)

such that ∣y(t)∣ ≤ r(t) and f(y(t)) ≤ t. In particular, there exists y such that ∣y∣ ≤

r(f(x)− ε(x)/2) and f(y) ≤ f(x)− ε(x)/2. If ∣x−w∣ < ω−1(f, ε/8) and ∣y − z∣ < ω−1(f, ε/8)


then by Corollary 4

∣w∣ − ∣z∣ ≥ (∣x∣ − ∣x −w∣) − (∣y∣ + ∣y − z∣)

≥ ω−1(f∗, ε/2) − 2ω−1(f, ε/8)

≥ ω−1(f∗, ε/2) − ω−1(f, ε/4)

≥ ω−1(f∗, ε/4)

and

f(w) − f(z) ≥ (f(x) − f(y)) − (∣f(x) − f(w)∣ + ∣f(y) − f(z)∣)

≥ ε/2 − ε/4

= ε/4.

The case f∗(x) − f(x) > 0 is treated similarily and will be ommited.

Proof of Theorem 4. For every f ∈ Cc(Rd), let f i(σσσ) = fσ1⋯σi and

I(f) = ∫Rd

∣x∣f(x)dx. (4.12)

By Proposition 1

EP [I (f i−1(σσσ) − f i(σσσ))]

equals

EP [∫Xσi±

[(f i−1(σσσ)(x) − f i−1(σσσ)(σi(x)))(∣x∣ − ∣σi(x)∣)]−dx] . (4.13)

By the independence of random variables Xi, 4.13 equals

EP [∫Ω∫Xσ±

[(f i−1(σσσ)(x) − f i−1(σσσ)(σ(x)))(∣x∣ − ∣σ(x)∣)]− dxdµi(σ)] . (4.14)


It is clear from Corollary 1 combined with Proposition 5 that

I(f∗) ≤ I(fσ) (4.15)

for all σ ∈ Ω and for all f ∈ Cc(Rd). This implies that

∞∑i=1

EP [I (f i−1(σσσ) − f i(σσσ))] = limn→∞

EP [I (f − fn(σσσ))]

≤ I(f − f∗)

< ∞

for all f ∈ Cc(Rd). In particular

∞∑i=1∫

Ω∫Xσ±

[(f i−1(σσσ)(x) − f i−1(σσσ)(σ(x)))(∣x∣ − ∣σ(x)∣)]− dxdµi(σ) <∞ (4.16)

almost surely, for all f ∈ Cc(Rd). Fix f ∈ Cc(Rd) and suppose that σσσ has the property

∥f i(σσσ) − f∗∥∞ ≥ ε > 0 for all i ≥ 1. This implies that there exists xi such that

∥f i(σσσ)(xi) − f∗(xi)∥∞ ≥ ε (4.17)

for all i ≥ 1. By Lemma 1, there exists a sequence yi such that

∣∣w∣ − ∣z∣∣ ≥ ω−1(f∗, ε(x)/4) (4.18)

and

(∣w∣ − ∣z∣)(f i−1(σσσ)(w) − f i−1(σσσ)(z)) ≥ ω−1(f∗, ε/4)ε/4 (4.19)

for every

(w, z) ∈ B(xi, ω−1(f i−1(σσσ), ε/8)) ×B (yi, ω−1(f i−1(σσσ), ε/8))

and for all i ≥ 1. Corollary 4 shows that polarization does not increase the modulus of


continuity and thus by 4.19

∫Ω∫Xσ±

[(f i−1(σσσ)(x) − f i−1(σσσ)(σ(x)))(∣x∣ − ∣σ(x)∣)]− dxdµi(σ)

is bounded below by

m (1/2ω−1(f, ε/8)Bd)ω−1(f∗, ε/4)ε/4 ⋅ µi (xi,B (yi,1/2ω−1(f, ε/8))) (4.20)

for all i ≥ 1. However, by the assumption 1.18 on the random variables Xi as well as 4.18:

∞∑i=1

µi (xi,B (yi,1/2ω−1(f, ε/8))) =∞. (4.21)

In particular, 4.16 is infinite. We have thus shown that for every fixed f ∈ Cc(Rd)

P ( limn→∞

∥fn(σσσ) − f∗∥∞ = 0) = 1. (4.22)

Let

fk∞k=1 ⊂ Cc(Rd) (4.23)

be a dense sequence (with respect to the L∞ norm) and let

A =∞⋂k=1

σσσ ∶ limn→∞

∥fn(σσσ)k − f∗k ∥∞ = 0 . (4.24)

If 1 ≤ p ≤ ∞ and f ∈ Lp(Rd) with f ∈ Cc(Rd) whenever p = ∞ then there exists a

subsequence kj∞j=1 such that

limj→∞

∥fkj − f∥p = 0. (4.25)


If σσσ ∈ A then by contraction property of rearrangements (Proposition 3)

∥fn(σσσ) − f∗∥p ≤ ∥fn(σσσ) − fn(σσσ)kj∥p + ∥fn(σσσ)kj

− f∗kj∥p + ∥f∗kj − f∗∥p

≤ 2∥fkj − f∥p + ∥fn(σσσ)kj− f∗kj∥p

for all (j, n) ∈ N ×N and thus

lim supn→∞

∥fn(σσσ) − f∗∥p ≤ 2∥fkj − f∥p (4.26)

for all j ≥ 1. Taking j to infinity on the right-hand side of 4.26 yields

limn→∞

∥fn(σσσ) − f∗∥p = 0. (4.27)

By 4.22

P (A) = 1 − P (∞⋃k=1

σσσ ∶ limn→∞

∥fn(σσσ)k − f∗k ∥∞ > 0)

≥ 1 −∞∑i=1

P ( limn→∞

∥fn(σσσ)k − f∗k ∥∞ > 0)

= 1.

4.3 Random Sequences of Steiner Symmetrizations

Proof of Theorem 5. We let

Si(uuu)(f) = Su1,...,ui(f) (4.28)

and

I(f) = ∫Rd

∣x∣f(x)dx (4.29)


for all f ∈ Cc(Rd). By the independence of the random variables Yi,

EP [I(Si−1(uuu)(f) − Si(uuu)(f))]

equals

EP [∫Sd−1

I (Si−1(uuu)(f) − Su(Si−1(uuu)(f)))dµi(u)] . (4.30)

However,

∞∑i=1

EP [I(Si−1(uuu)(f) − Si(uuu)(f))] = limn→∞

EP [I(f − Sn(uuu)(f))]

≤ I(f − f∗)

< ∞

and thus

EP [∞∑i=1∫Sd−1

I (Si−1(uuu)(f) − Su(Si−1(uuu)(f)))dµi(u)] <∞.

In particular,∞∑i=1∫Sd−1

I (Si−1(uuu)(f) − Su(Si−1(uuu)(f)))dµi(u) <∞ (4.31)

almost surely. If uuu satisfies ∥Si(uuu)(f)−f∗∥∞ ≥ ε > 0 for all i ≥ 1 then by Lemma 1 combined

with the fact that Steiner symmetrization does not increase the modulus of continuity

(see Proposition 8), there exists a sequence of pairs (xi, yi) ⊂ Rd ×Rd such that

∣∣w∣ − ∣z∣∣ ≥ ω−1(f∗, ε(x)/4) (4.32)

and

(∣w∣ − ∣z∣)(Si−1(uuu)(f)(w) − Si−1(uuu)(f)(z)) ≥ ω−1(f∗, ε/4)ε/4 (4.33)


for every

(w, z) ∈ B(xi, ω−1(f, ε/8)) ×B (yi, ω−1(f, ε/8))

⊂ B(xi, ω−1(Si−1(uuu)(f), ε/8)) ×B (yi, ω−1(Si−1(uuu)(f), ε/8)) .

By Proposition 8,

I (g − Su(f)) ≥ I (g − gσ0,ru) (4.34)

for all r ∈ R − 0, for all u ∈ Sd−1 and for all g ∈ Cc(Rd). In particular, by using the

polarization formula for functionals (Proposition 1), we immediately see from 4.33 and

4.34 that

∫Sd−1

I (Si−1(uuu)(f) − Su(Si−1(uuu)(f)))dµi(u)

is bounded below by

m (1/2ω−1(f, ε/8)Bd)ω−1(f∗, ε/4)ε/4 ⋅ µi(Ai) (4.35)

with

Ai = xi − z∣xi − z∣

∶ z ∈ B(yi,1/2ω−1(f, ε/8)) .

However, by the assumption 1.21 on the random variables Yi as well as 4.32, one has

∞∑i=1

µi(Ai) =∞ (4.36)

and thus the left-hand side of 4.31 is infinite. The probability that the left hand side of

4.31 is infinite is zero and thus we may finally conclude that

P ( limn→∞

∥Sn(uuu)(f) − f∗∥∞ = 0) = 1 (4.37)


for all f ∈ Cc(Rd). Mimicking the proof of Theorem 4, we see that 4.37 implies Theorem

5.

4.4 Examples

We first show that Theorems 4 and 5 are extensions of Theorem 3 by showing that if the

conditions 1.9 and 1.10 of Theorem 3 are satisfied then so are the conditions 1.18 and

1.21 of Theorems 4 and 5

Proposition 12. If Xi is a sequence of independent random variables such that

lim infi→∞

P (Xi ∈ Bρ(σ,λ)) > 0 (4.38)

for all σ ∈ Ω and λ > 0 then for any λ1 > 0

∞∑i=1

µi(xi,B(yi, λ1)) =∞ (4.39)

for every bounded sequence (xi, yi)∞i=1 ⊂ Rd ×Rd.

Proof. Let (xi, yi)∞i=1 be any bounded sequence and let λ1 > 0. Let ik be a subsequence

such that (xik , yik) converges to a pair (x, y). For sufficiently large K

σ ∶ σ(x) ∈ B(y, λ1/2) ⊂ σ ∶ σ(xik) ∈ B(yik , λ1) (4.40)

for all k ≥ K. Recalling that σx,z denotes the unique reflection that maps x to z, it is

easily checked that

σx,z(w) = (∣x −w∣2 − ∣z −w∣2)(x − z)∣x − z∣2

(4.41)

for all x, z,w. In particular, the map z ↦ σx,z(0) = w is continuous on the open set Rd−x

with continuous inverse σ0,w(x). As a consequence, the set σ ∶ σ(x) ∈ B(y, λ1/2) has a


non-empty interior with respect to the metric ρ on Ω. By the assumption 4.38, we must

have

lim infk→∞

µik(x,B(y, λ1/2)) > 0. (4.42)

Combining 4.40 and 4.42, we conclude that

∞∑i=1

µi(x,B(yi, λ1) =∞. (4.43)

It is clear that a similar (and easier) proof works for sequences of independent random

variables Yi distributed in Sd−1:

Proposition 13. If Yi is a sequence of independent random variables distributed in Sd−1

such that

lim infn→∞

µn (B(u,λ)) > 0 (4.44)

for every u ∈ Sd−1 and for every λ > 0 then

∞∑i=1

µi (B(ui, λ)) =∞ (4.45)

for every sequence ui∞i=1 ⊂ Sd−1 and for every λ > 0.

We wish to give an example of a sequence of independent random variables Xi dis-

tributed in Ω such that the condition 1.18 holds but not 4.38. To do so, we will need the

following Lemma:

Lemma 2. If Tx(y) denotes the transformation T (y) = σx,y(0) then the Jacobian JTx(y)

equals

(∣∣x∣2 − ∣y∣2∣∣x − y∣2

)d−1

. (4.46)


Proof. Let y ∈ Rd with ∣y∣ ≠ ∣x∣ and let 0 < ε < ∣∣x∣ − ∣y∣∣. By using polar coordinates

(y = x + ru):

m(B(y, ε)) = γd∫Sd−1

∫b(x,u,ε)

a(x,u,ε)rd−1drdθ(u) (4.47)

with [a(x,u, ε), b(x,u, ε)] = r ∶ x + ru ∈ B(y, ε). However, the image of the line seg-

ment x + ru ∶ r ∈ [a(x,u, ε), b(x,u, ε)] is the line segment Tx(x + a(x,u, ε)) + r ∶ r ∈

[0, b(x,u, ε) − a(x,u, ε)] and consequently by using polar coordinates (y = ru)

m(Tx(B(y, ε))) = γd∫Sd−1

∫∣T (x+a(x,u,ε))∣+c(x,u,ε)

∣T (x+a(x,u,ε))∣rd−1drdθ(u) (4.48)

with c(x,u, ε) = ±(b(x,u, ε) − a(x,u, ε)) depending on the sign of ⟨u,x⟩. Clearly

limε→0

∫∣T (x+a(x,u,ε))∣+c(x,u,ε)∣T (x+a(x,u,ε))∣ rd−1dr

∫b(x,u)a(x,u) r

d−1dr= (∣Tx(y)∣

∣x − y∣)d−1

(4.49)

and since

σx,y(0) =(∣x∣2 − ∣y∣2)(x − y)

∣x − y∣2(4.50)

then

∣Tx(y)∣∣x − y∣

= ∣∣x∣2 − ∣y∣2∣∣x − y∣2

. (4.51)

For any set A ∈ F , we denote by A0 the set σ(0) ∶ σ ∈ A and we let φ(y) denote the

standard Gaussian:

φ(y) = (2π)−d/2e−∣y∣2/2. (4.52)

We define the probability measures

µi(A) = αdi ∫A0φ(αiy)dy (4.53)

with αi positive constants to be determined. Let xi∞i=1 be any bounded sequence con-


tained in Rd and let 0 < λ < ∣xi∣ ≤ λ1 for all i ≥ 1. If Bi ⊂ (∣xi∣ − λ)Bd for all i ≥ 1 is a

sequence of balls of uniform radius then by Lemma 2

µi(xi,Bi) = αdi ∫Bi

(∣∣xi∣2 − ∣y∣2∣∣xi − y∣2

)d−1

φ(αiy)dy. (4.54)

However, we have

∣∣xi∣2 − ∣y∣2∣∣xi − y∣2

≥ ∣∣xi∣ − ∣y∣∣∣xi∣ + ∣y∣

≥ λ

2λ1

(4.55)

for all y ∈ Bi and for all i ≥ 1. In addition,

φ(αiy) ≥ (2π)−d/2αdi (eα2i )−λ2/2

for all y ∈ Bi and for all i ≥ 1. As a result there exists a constant c not depending on i

such that

µi(xi,Bi) ≥ cαdi (eα2i )−λ2/2 (4.56)

for all i ≥ 1. Since we want our our measures µi to satisfy condition 1.18, we need to

choose αi so that∞∑i=1

αdi (eα2i )−λ2/2 =∞. (4.57)

Letting αi =√

log log(i) for i ≥ 3 yields 4.57. On the other hand, it is clear that

limi→∞

µi(Bρ(σ, ∣σ(0)∣/2)) = limi→∞

αdi ∫B(σ(0),∣σ(0)∣/2)

φ(αiy)dy = 0 (4.58)

for every σ ∈ Ω. In other words, if Xi is a sequence of independent random variables with

P (Xi ∈ A) = µi(A) for every A ∈ F and for every i ≥ 1 then the condition 1.18 is satisfied

but not the condition 4.38.

As for sequence of polarizations, we will try to construct a sequence of random vari-

ables Yi distributed in Sd−1 that satisfy the condition 1.21 but not the condition 4.44.


Letting

µi(B) = ∫Sd−1

1 − ∣xi∣2∣xi − u∣d

dθ(u) (4.59)

with xi ∈ Rd and ∣xi∣ < 1 for all i ≥ 1 defines a sequence of probability measures. In fact, µi

is the “hitting distribution ” on Sd−1 for Brownian motion starting at xi (see [13, p.102]).

Let ui be any sequence on Sd−1 and let λ > 0. We have

∞∑i=1

µi(B(ui, λ)) = ∫B(ui,λ)

1 − ∣xi∣2∣xi − u∣d

dθ(u) (4.60)

≥ 2−(d−1)∞∑i=1

θ(B(ui, λ) ∩ Sd−1)(1 − ∣xi∣). (4.61)

Since θ(B(ui, λ)∩Sd−1) is equal for all i then choosing xi = (1−1/i,0, . . . ,0) yields that the

right-hand side of 4.61 diverges and consequently condition 1.21 is satisfied. However, µi

converges weakly to a point mass at (1,0, . . . ,0) (see [13, p.31]) and so certainly condition

4.26 is not satisfied. In other words, if Yi is a sequence of independent random variables

with distribution µi then condition 1.21 is satisfied but condition 4.26 is not satisfied.

Chapter 5

Rates of Convergence: Holder

Continuous Functions

5.1 Polarization of Holder Continuous Functions

Let f ∈ C0,α(λBd) with λ > 0 and let

φ(y) = χ∣y∣≤2λ(y)(2λγd)−1∣y∣−(d−1). (5.1)

For any A ∈ F , we let

µ(A) = ∫A0φ(y)dy (5.2)

with A0 = σ(0) ∶ σ ∈ A. By construction, µ is a probability measure on Ω. If we let

I(f) = ∫Rd

∣x∣f(x)dx (5.3)

then the following holds:

46

Chapter 5. Rates of Convergence: Holder Continuous Functions 47

Lemma 3. Eµ [I(f − fσ)] is bounded below by

2−(d−1)∫f>f∗

∣x∣−(d−1)∫∣y∣<∣x∣

(f(x) − f(y))−(∣x∣ − ∣y∣)dφ(y)dydx.

Proof. By the polarization formula for functionals (Proposition 1):

Eµ [I(f − fσ)] = ∫Ω∫Xσ−

(∣x∣ − ∣σ(x)∣)(f(x) − f(σ(x)))−dxdP (σ). (5.4)

By Fubini’s theorem, the right-hand side of 5.4 equals

∫λBd

∫∣σ(x)∣<∣x∣

(∣x∣ − ∣σ(x)∣)(f(x) − f(σ(x)))−dP (σ)dx. (5.5)

By letting y = σ(x) and using Lemma 2, 5.5 equals

∫λBd

∫∣y∣<∣x∣

(∣x∣ − ∣y∣)(f(x) − f(y))− (∣x∣2 − ∣y∣2∣x − y∣2

)d−1

φ(y)dydx. (5.6)

To complete the proof, note that the triangle inequality yields

∣x∣2 − ∣y∣2∣x − y∣2

≥ ∣x∣ − ∣y∣∣x∣ + ∣y∣

≥ ∣x∣ − ∣y∣2∣x∣

(5.7)

for ∣y∣ < ∣x∣.

Lemma 4. If f(x) > f∗(x) then

∫∣y∣<∣x∣

(f(x) − f(y))− (∣x∣ − ∣y∣)dφ(y)dy ≥ ∫∣y∣<∣x∣

(f(x) − f∗(y))− (∣x∣ − ∣y∣)dφ(y)dy. (5.8)

Proof. Fix x such that f(x) > f∗(x) and let φ1(y) = (∣x∣ − ∣y∣)dφ(y) then it follows from

the proof of the Hardy-Littlewood inequality that it suffices to show that

m(y ∶ ∣y∣ < ∣x∣, f(y) < f(x) − t, φ1(y) > s) (5.9)


is greater or equal to

m(y ∶ ∣y∣ < ∣x∣, f∗(y) < f(x) − t, φ1(y) > s) (5.10)

for every 0 ≤ t < f(x) and s ≥ 0. From the proof of Lemma 1 (see 4.7), if 0 ≤ t < f(x)

then

m(y ∶ f(y) < f(x) − t, ∣y∣ < ∣x∣) ≥m(y ∶ f∗(y) < f(x) − t, ∣y∣ < ∣x∣) (5.11)

for every 0 ≤ t < f(x). Since y ∶ φ(y) > s, ∣y∣ < ∣x∣ is an open ball centered at the origin,

then it is clear from 5.11 that 5.9 is greater or equal to 5.10.

To find a lower bound on the right-hand side of 5.8 in terms of f(x) − f∗(x), we will

need the following simple Lemma:

Lemma 5. [8, p.137]If h1(r), h2(r) are non-negative increasing and decreasing functions

respectively with h1(a) = 0 and h2(b) = 0 then

supr∈(a,b)

h1(r)h2(r) ≤ ∫b

a−h1(t)h′2(t)dt.

Proof.

supr∈(a,b)

h1(r)h2(r) = supr∈(a,b)

h1(r)∫b

r−h′2(t)dt

≤ supr∈(a,b)

∫b

r−h1(t)h′2(t)dt

= ∫b

a−h1(t)h′2(t)dt.

Proposition 14.

Eµ [I(f − fσ)] ≥ γ− d+1α

d C3(α, d, [f]α, λ)∥f − f∗∥1+ d+1

α1


with C3(α, d, [f]α, λ) equal to

[f]−(d+1)/αα λ−d(1+ d+1

α)2−(d+1)(1+1/α)α

(d + 1 + α)2e(1/d)− d+1α .

Proof. By Lemmas 3 and 4, Eµ[I(f − fσ)] is bounded below by

λ−12−d∫f>f∗

∣x∣−(d−1)γ−1d ∫

∣y∣<∣x∣(f(x) − f∗(y))−(∣x∣ − ∣y∣)d∣y∣−(d−1)dydx. (5.12)

Applying Lemma 5 to the special case

h1(r) = (f(x) − f∗(r)), h2(r) = (d + 1)−1(∣x∣ − r)d+1

shows that

γ−1d ∫

∣y∣<∣x∣(f(x) − f∗(y))−(∣x∣ − ∣y∣)d∣y∣−(d−1)dy = ∫

∣x∣

r(f(x))(f(x) − f∗(r))(∣x∣ − r)ddr

is bounded below by

supr∈(r(f(x)),∣x∣)

(d + 1)−1(f(x) − f∗(r))(∣x∣ − r)d+1. (5.13)

It is clear that 5.13 also equals

(d + 1)−1 supt∈(f∗(x),f(x))

(f(x) − t)(∣x∣ − r(t))d+1. (5.14)

Since f ∈ C0,α(Bd) and the decreasing symmetric rearrangement does not increase the

modulus of continuity (see Corollary 4) then f∗ ∈ C0,α(λBd) with [f∗]α ≤ [f]α and

∣x∣ − r(t) ≥ [f∗]−1/αα (t − f∗(x))1/α. This implies that 5.14 is bounded below by

(d + 1)−1[f∗]−(d+1)/αα sup

t∈(f∗(x),f(x))(t − f∗(x)) d+1α (f(x) − t) (5.15)


which equals

(d + 1)−1 (d + 1

α)d+1α

(1 + d + 1

α)−(1+ d+1

α)[f∗]−(d+1)/α

α (f(x) − f∗(x)) d+1α +1. (5.16)

Noting that

(d + 1)−1 (d + 1

α)d+1α

(1 + d + 1

α)−(1+ d+1

α)≥ e−1α

(d + 1 + α)2(5.17)

gives

Eµ [I(f − fσ)] ≥ C1(α, d, [f∗]α, λ)∫f>f∗

∣x∣−(d−1)(f(x) − f∗(x))1+ d+1α dx (5.18)

with

C1(α, d, [f∗]α, λ) = λ−1[f∗]−(d+1)/αα 2−d

e−1α

(d + 1 + α)2. (5.19)

Applying the reverse Holder inequality [4, p.225] with parameters p = (1+ d+1α )−1 and

q = − αd+1 yields

(5.20)

∫f>f∗

∣x∣−(d−1)(f(x) − f∗(x))1+ d+1α dx

is bounded below by

(∫f>f∗

∣x∣α(d−1)d+1 dx)

− d+1α

(∫f>f∗

(f(x) − f∗(x))dx)1+ d+1

α

. (5.21)

From the identity

∫f>f∗

f(x)dx + ∫f<f∗

f(x)dx = ∫f>f∗

f∗dx + ∫f<f∗

f∗dx

we deduce that

2∫f>f∗

(f − f∗)dx = ∥f − f∗∥1. (5.22)


By 5.18,5.21 and 5.22, we finally obtain

Eµ [I(f − fσ)] ≥ γ− d+1α

d C2(α, d, [f∗]α, λ)∥f − f∗∥1+ d+1

α1 (5.23)

with C2(α, d, [f∗]α, λ) equal to

[f∗]−(d+1)/αα

2−(d+1)(1+1/α)λ−d(1+ d+1α

)α

(d + 1 + α)2e( (d + 1)α(d − 1) + d(d + 1)

)− d+1α

. (5.24)

Following section 4.1 (“Random Variables Distributed in Ω”), there exists a unique

probability measure P defined on the infinite product space (ΩN,FN) such that

P (σ1 ∈ A1, . . . , σn ∈ An) =n

∏i=1

µ(Ai). (5.25)

for every n ≥ 1. We also let

fn(σσσ) = fσ1⋯σn (5.26)

for all σσσ ∈ ΩN.

Proposition 15.

EP [(γ−1d ∥fT (σσσ) − f∗∥1)

1+ d+1α ] ≤ (T (d + 1))−1λd+1C3(α, d, [f∗]α, λ)−1∥f − f∗∥∞

for all T ≥ 1.

Proof. We have

EP [I (f − fT (σσσ))] =T

∑n=1

EP [I (fn−1(σσσ) − fn(σσσ))] (5.27)

for all T ≥ 1. By Proposition 14 as well as the contraction property of rearrangements


(Proposition 3):

EP [I (fn−1(σσσ) − fn(σσσ))] ≥ γ−d+1α

d C3(α, d, [f∗]α, λ)EP [∥fT (σσσ) − f∗∥1+ d+1α

1 ] (5.28)

for all n = 1, . . . , T . Using 5.27, we see that 5.28 yields

EP [I (f − fT (σσσ))] ≥ Tγ−d+1α

d C3(α, d, [f]α, λ)EP [∥fT (σσσ) − f∗∥1+ d+1α

1 ] (5.29)

for all T ≥ 1. However, we also have (see 4.15)

EP [I (f − fT (σσσ))] ≤ I(f − f∗) ≤ (d + 1)−1λd+1γd∥f − f∗∥∞. (5.30)

Combining 5.29 and 5.30 completes the proof.

Lemma 6. If g ∈ C0,α(Rd) ∩Cc(Rd) then

[g]−d/α2

αα

d(α + d)∥g∥1+d/α

∞ ≤ γ−1d ∥g∥1 (5.31)

Proof. Let g ∈ C0,α(Rd) ∩Cc(Rd) with [g]α = 1. Let x1 be any point where ∣g∣ attains a

maximum. By using polar coordinates (x = x1 + ru), we have

γ−1d ∥g∥1 = ∫

Sd−1∫

∞

0∣g(ru)∣rd−1drdθ(u) (5.32)

. In particular, there exists u1 such that

∫∞

0∣g(ru1)∣rd−1dr ≤ γ−1

d ∥g∥1. (5.33)

Since g belongs to C0,α with [g]α = 1 then

∣g(ru1)∣ ≥ ∥g∥∞ − rα (5.34)


for all r ∈ (0, ∥g∥1/α∞ ) and thus

∫∥g∥1/α∞

0(∥g∥∞ − rα)rd−1dr = α

d(α + d)∥g∥1+d/α

∞ ≤ γ−1d ∥g∥1. (5.35)

This proves the Lemma for the case [g]α = 1 and the general case follows by scaling.

Proposition 16. There exists an explicit constant C(α, d, [f]α, λ) such that

EP [∥fT (σσσ) − f∗∥∞] ≤ ε (5.36)

whenever

T ≥ C(α, d, [f]α, λ)(1/ε)(1+ d+1α

)(1+ dα)∥f − f∗∥∞ (5.37)

for all ε > 0 and for all f ∈ C0,α(λBd).

Proof. By Lemma 6 and Proposition 15 combined with the fact that

[f∗]α ≤ [fσ]α ≤ [f]α (5.38)

for all σ ∈ Ω, we have

EP [∥fT (σσσ) − f∗∥(1+dα)(1+ d+1

α)

∞ ] ≤ T −1C(α, d, [f]α, λ)∥f − f∗∥∞

with

C(α, d, [f]α, λ) =(α + d)d(d + 1)α

(2[f]α)d/α2

λd+1C3(α, d, [f]α, λ)−1.

By Jensen’s inequality

EP [∥fT (σσσ) − f∗∥(1+dα)(1+ d+1

α)

∞ ] ≥ EP [∥fT (σσσ) − f∗∥∞](1+dα)(1+ d+1

α)


and thus

EP [∥fT (σσσ) − f∗∥∞] ≤ (T −1C(α, d, [f]α, λ)∥f − f∗∥∞)[(1+dα)(1+ d+1

α)]−1

.

In particular, if

T ≥ C(α, d, [f]α, λ)∥f − f∗∥∞(1/ε)(1+ dα )(1+ d+1α

)

then

EP [∥fT (σσσ) − f∗∥∞] ≤ ε.

However, a computation shows that C(α, d, [f]α, λ) equals

(1/d) d+1α e(α + d)d(d + 1)α2

[f]d+1α+d/α2

α λ2d+1+ d(d+1)α 2(d+1)(1+1/α)+ d

α2 . (5.39)

We make a quick remark regarding the constant C(α, d, [f]α, λ). It is clear that there

exists a numerical constant c such that

e(α + d)d(d + 1)α2

2(d+1)(1+1/α)+ dα2 ≤ 2cd/α

2

and

[f]d+1α+d/α2

α ≤ [f]cd/α2

α .

In particular, we see that there exists a numerical constant c such that

C(α, d, [f]α, λ) ≤ (2[f]α)cd/α2

λ2d+1+ d(d+1)α (1/d) d+1α . (5.40)

To complete this section, we note that Theorem 7 is an immediate corollary to Propo-

sition 16.


5.2 Steiner Symmetrization of Holder Continuous Func-

tions

We will now use the results of the previous section to give a quick proof of Theorem 8.

Following section 4.1 (“Random Variables Distributed in Sd−1”), there exists a unique

probability measure P ′ defined on the infinite product space ((Sd−1)N,GN) such that

P ′(u1 ∈ B1, . . . , un ∈ Bn) =n

∏i=1

θ(Bi). (5.41)

for every n ≥ 1. If we let

Sn(uuu)(f) = Su1,...,un(f) (5.42)

for all n ≥ 1 and for all uuu ∈ ΩN then we have the following proposition:

Proposition 17.

EP ′ [∥ST (uuu)(f) − f∗∥∞] ≤ ε (5.43)

whenever

T ≥ C(α, d, [f]α, λ)(1/ε)(1+ d+1α

)(1+ dα)∥f − f∗∥∞ (5.44)

for all ε > 0 and for all f ∈ C0,α(λBd).

Proof. Let f ∈ C0,α(λBd) for some λ > 0. It is clear from Proposition 8 that

I(f − fσ0,ru) ≤ I(f − Su(f)) (5.45)

for all r ∈ Rd and for all u ∈ Sd−1. However, using the notation of the previous section


(see 5.2) and 5.45, we obtain

Eµ [I(f − fσ] = ∫2λBd

I(f − fσ0,y)φ(y)dy

= ∫Sd−1

1/2λ∫2λ

0I(f − fσ0,ru)drdθ(u)

≤ ∫Sd−1

I(f − Su(f))dθ(u)

= Eθ [I(f − Su(f))] .

By Proposition 14:

Eθ [I(f − Su(f))] ≥ γ− d+1α

d C3(α, d, [f]α, λ)∥f − f∗∥1+ d+1

α1 . (5.46)

This shows that the analog of Proposition 14 for Steiner symmetrization also holds. To

prove Proposition 15, all that was needed was Proposition 14 as well as the contraction

property (Proposition 3) of rearrangements. Hence, Proposition 15 also holds for Steiner

symmetrization with P replaced by P ′. The only property of polarization that was needed

to prove Proposition 16 from Proposition 15 was

[f∗]α ≤ [fσ]α ≤ [f]α (5.47)

for all σ ∈ Ω. However, by Proposition 8, the analogous statement for Steiner symmetriza-

tion also holds i.e.,

[f∗]α ≤ [Su(f)]α ≤ [f]α (5.48)

for all u ∈ Sd−1.

To completes this section, we note that Theorem 8 is an immediate corollary to

Proposition 17.

Chapter 6

Rates of Convergence: Convex

Bodies

6.1 Preliminaries

6.1.1 Chapter Outline

We have seen in chapter 4 that almost every random sequence of Steiner symmetrizations

will transform any arbitrary compact set F into F ∗. In this chapter, we study how fast

the transformation can occur when A is convex. More precisely, given ε > 0, what is

the minimal number of Steiner symmetrizations needed to transform any convex body

K with volume κd into a new convex body K ′ with the property

(1 − ε)Bd ⊂K ′ ⊂ (1 + ε)Bd?

As described in the introduction, the purpose of this chapter is to give a self-contained

derivation of Theorem 6 due to Klartag. It should be stressed that the proof Theorem 6

depends fundamentally on a previous result due to Klartag and Milman [10]:

Theorem 9. There exists 3d Steiner symmetrizations that transform any convex body K

57

Chapter 6. Rates of Convergence: Convex Bodies 58

with volume κd into an isomorphic ball i.e., there exists 3d Steiner symmetrizations that

transform any initial convex body K with volume κd into a new convex body K ′ with the

property

c1Bd ⊂K ′ ⊂ c2B

d (6.1)

for some numerical constants c1 and c2.

6.1.2 Notation and Preliminary Facts

In what follows, the symbol Eν(⋅) will always refer to expectation with respect to the

probability measure ν defined on some contextual space and Vn will refer to the n di-

mensional Hausdorff measure. We also let

h(K,x) = supk∈K

⟨x, k⟩ (6.2)

and

b(K) = 2 ⋅ ∫Sd−1

h(K,u)dθ(u) (6.3)

denote the support function and mean width of K respectively. For any two convex bodies

K and L, we have the identity [15, p.53]

∥h(K,x) − h(L,x)∥∞ = δ(K,L). (6.4)

We let

R(K) = infr ∶K ⊂ rBd (6.5)

and

r(K) = supr ∶K ⊂ rBd (6.6)

denote the circumradius and inradius of K respectively.

If u ∈ Sd−1 then we define the Minkowski symmetrization of K with respect to u by


Mu(K) = K+πu(K)2 with πu denoting the reflection across the orthogonal complement of

u. It is clear from the definition that Minkowski symmetrizaton preserves mean width

and always contains Steiner symmetrization i.e., for all u ∈ Sd−1

Su(K) ⊂Mu(K). (6.7)

We first consider the effect of d Minkowski symmetrizations on a convex body K

with respect to the d orthonormal column vectors of an arbitrary orthonormal matrix.

Consider the discrete cube ±1d equipped with the uniform probability measure ν and

given f in L2(Sd−1) define

fρ(u) = Eν [f (d

∑i=1

εi ⟨u, ρi⟩ρi)] (6.8)

with ρ = (ρ1, . . . , ρd) an element of Od - the group of orthogonal matrices. We de-

fine Sρd to be the subspace of L2(Sd−1) of functions f that have the property f(u) =

f (∑di=1 εi ⟨u, ρi⟩ρi) for all ε ∈ ±1d.

Proposition 18. For all f in L2(Sd−1), projSρd(f) = fρ. In addition, if Mρ(K) is the

convex body obtained from applying d Minkowski symmetrizations with respect to the

column vectors of ρ to the convex body K then h(Mρ(K), u) = hρ(K,u).

Proof. Given f ∈ L2(Sd−1) and v ∈ Sd−1, f v(u) = 1/2[f(u) + f(πv(u))] is the projec-

tion of f onto the subspace of L2(Sd−1) consisting of functions that are invariant with

respect to the reflection πv. We then obtain projSρd(f) = f e1⋯ed = fρ. To prove the

second statement of the proposition, note that for any convex body K, h(MK(v), u) =

1/2[h(K,u) + h(K,πv(u))] and thus h(Kρ, u) = hρ1⋯ρd(K,u) = hρ(K,u).


6.1.3 Outline of the Strategy

We now give a brief outline of the strategy that was used by Klartag to derive Theorem

6.

The first step is to show that applying Minkowski symmetrizations with respect to

the column vectors of random orthogonal matrices will (on average) transform any initial

isomorphic ball into a ball of equal mean width. Using results from the theory of spherical

harmonics, it is shown that the convergence described in the previous sentence occurs (on

average) exponentially fast. This is done in section 6.2 (“Minkowski Symmetrizatons”).

The second step consists of making the link between Minkowski symmetrization and

Steiner symmetrization. It is clear that given any convex body its circumradius always

bound half its mean width. In fact, one can give a quantitative estimate on the relative

size of the circumradius and half the mean width of convex bodies of volume κd. The

estimate shows that half the mean width is strictly less than circumradius and quanti-

fies the difference. The derivation of this estimate is done in section 6.3 (“Steiner and

Minkowski Symmetrization: Making the Link”) .

In the third step, Theorem 9 is used and 3d Steiner symmetrizations are used to

transform any convex body K with volume κd into an isomorphic ball (see 6.1).

In the fourth step, the fact that Minkowski symmetrization always contains Steiner

symmetrization as well as step 2 is used to deduce that applying Steiner symmetriza-

tions with respect to the column vectors of random orthogonal matrices will (on average)

decrease circumradius. The expected decrease is quantified by using the rates of con-

vergence for Minkowski symmetrizations found in step 1. It is shown that (on average)

the circumradius decays exponentially fast. This is done in subsection 6.4.1 (“Circumra-

dius”).

In the final step, the relationship between circumradius and inradius is considered. It

is shown that for convex bodies of volume κd, the inradius depends on the circumradius

in the sense that if the circumradius is small then the inradius is large (see subsection


6.4.2 (“Inradius”)). By using the fourth step, one can finally obtain Theorem 6 (see

section 6.4.3 (“Final Results”)).

6.2 Minkowski Symmetrizations

6.2.1 Notation and Preliminary Facts

We let ETµ(⋅) denote the expectation with respect to the probability measure µ × ⋯ × µ

(T times) defined on the product space Od ×⋯ ×Od (T times). An arbitrary element of

Od ×⋯ ×Od (T times) will be denoted by ρρρ. We suppress the dependence on T since it

will be clear from the context. If (ρ1, . . . , ρT ) = ρρρ then for any f ∈ L2(Sd−1), we let (recall

6.8)

fρρρ ∶= fρ1⋯ρT .

We will refer to Hnd as the space of spherical harmonics of degree n - the restriction to

Sd−1 of harmonic homogeneous polynomials of degree n. The dimension of the vector

space Hnd will be denoted by N(d,n). If H(u) is a spherical harmonic, we will denote

by H(x) the unique harmonic polynomial defined on all of Rd whose restriction is H(u).

The inner product ⟨⋅, ⋅⟩ refers to the integral of the product of functions on Sd−1 with

respect to the measure θ. Finally, µ will denote the normalized Haar measure on Od and

O+d will denote the group of rotations.

Let K be any convex body with R(K) ≤ c with c > 0 arbitrary. It is shown in [12]

that if f is a continuous function on Sd−1 with modulus of continuity

ω1(f, s) = sup∣f(u1) − f(u2)∣ ∶ arccos(⟨u1, u2⟩) ≤ s (6.9)

then there exists a polynomial Pn(u) of degree at most n and a constant c3 not depending

on d nor n such that

∥f − Pn∥∞ ≤ c3ω1(f,1/n). (6.10)


Since Euclidean distance is bounded by geodesic distance:

ω1 (h(K,u), s) ≤ cs (6.11)

for all s > 0. By 6.10, given ε′ > 0, there exists a polynomial Pε′(K,u) of degree ⌈ dε′ ⌉ such

that

∥h(K,u) − Pε′(K,u)∥∞ < cc3ε′. (6.12)

It is well known that every polynomial can be written as a sum of spherical harmonics [7,

p.70]. More precisely, if Hij(u)N(d,j)i=1 denotes an orthonormal basis for Hjd then we have

Pε′(K,u) =⌈ dε′ ⌉

∑j=0

Qj(u) =⌈ dε′ ⌉

∑j=0

N(d,j)

∑i=1

⟨Pε′ ,Hij⟩Hij(u) (6.13)

and in particular

Q0(u) = ∫Sd−1

Pε′(u)dθ(u). (6.14)

To simplify the notation, we will temporarily let ⌈ dε′ ⌉ = n. We have by 6.12,6.13 and 6.14

combined with the triangle inequality

ETµ [∥hρρρ(K,u) − b(K)2

∥∞] ≤ 2cc3ε

′ +ETµ [∥Pρρρε′(K,u) −Q0(u)∥∞] (6.15)

≤ 2cc3ε′ +

n

∑j=1

ETµ [∥Qρρρj∥∞] . (6.16)

Recalling the identity 6.4 and using Proposition 18 as well as 6.16, we obtain the

geometric inequality:

ETµ [δ (Mρρρ(K), b(K)2

Bd)] ≤ 2cc3ε′ +

n

∑j=1

ETµ [∥Qρρρj∥∞] (6.17)

with n = ⌈ dε′ ⌉. However, by Proposition 18, we know that Qρρρj is the projection of Qj onto


the subspace Hjd ∩ Sρ1d ∩⋯ ∩ Sρdd . In particular, it is clear that

ETµ [∥Qρρρj∥2] < ∥Qj∥2 (6.18)

for all j ≥ 1. In fact, it will be shown using results from the theory spherical harmonics

that the left-hand side of 6.18 decays exponentially with respect to T (see subsection

6.2.3 “L2 Decay of Spherical Harmonics”). From the exponential L2 decay of spherical

harmonics (as described in the previous sentence), we will be able to conclude that for

fixed ε′, the whole expressionn

∑j=1

ETµ [∥Qρρρj∥∞] (6.19)

decays exponentially with respect to T (see subsection 6.2.4 “L∞ Decay of Spherical

Harmonics”). In the last subsection 6.2.5 (“Final Results”), we use the results described

in the previous sentence to finally conclude that the left-hand side of 6.17 decays expo-

nentially with respect to T .

6.2.2 Tools from Spherical Harmonics

We now present a few results from the theory of spherical harmonics that will be used in

subsections 6.2.3 and 6.2.4. The presentation is largely based on Chapter 3 of [7] with a

few changes.

Proposition 19. Let d ≥ 3 and let H be a nonzero linear subspace of Hnd with the

invariance property H(ρ(u)) ∈ H for all H ∈ H and for all ρ ∈ O+d . There exists a

function Q(t) such that for any orthonormal basis H1, . . . ,Hm of H and for any two unit

vectors u and v, ∑mi=1Hi(u)Hi(v) = Q(⟨u, v⟩).

Proof. Let H1, . . . ,Hm be any orthonormal basis for H then for every H ∈ H, we have

H(v) =m

∑i=1

⟨H(u),Hi(u)⟩Hi(v) = ⟨H(u),m

∑i=1

Hi(u)Hi(v)⟩ .


This shows that Q1(u, v) ∶= ∑mi=1Hi(u)Hi(v) is invariant under change of basis (this

holds for any d ≥ 2). To finish the proof, we need to show that Q1(u, v) depends only on

⟨u, v⟩ and not on the relative positions of u and v. If we have two pairs of unit vectors

say (u1, v1), (u2, v2) such that ⟨u1, v1⟩ = ⟨u2, v2⟩ then there exists a rotation ρ such that

ρ(u1) = u2, ρ(v1) = v2. Since H is invariant under rotations, H1 ρ, . . . ,Hm ρ is also a

basis for H and thus Q1(u1, v1) = Q1(ρ(u), ρ(v)) = Q1(u2, v2) for all ρ. In particular, we

have Q1(u, v) = Q1 ((0, . . . ,0), (0, . . . ,√

1 − t2, t)) ∶= Q(t) whenever ⟨u, v⟩ = t.

Lemma 7. Let x = (x1, . . . , xd−1,0) and let Tn(x) = ∑⌊n/2⌋i=0 a2ixn−2i

d ∣x∣2i with a2(i+1) =

a2i ⋅ (n−2i)(n−2i−1)(2i+2)(d+2i−1) and a0 = 1. H ∈ Hnd has the property H(ρ(u)) =H(u) for all rotations ρ

that fix ed if and only if H(u) is a constant multiple of Tn(u).

Proof. Let H(u) ∈ Hnd have the properties stated in the lemma. Clearly there exists

homogeneous polynomials pi of degree i and real numbers bi such that

H(x) =n

∑i=0

bixn−id pi(x).

Let ρ be a rotation leaving ed fixed and mapping x to (∣x∣,0, . . . ,0) then H(x) =H(ρ(x))

implies

H(x) =n

∑i=0

bi ⋅ xn−id pi(∣x∣) =⌊n/2⌋

∑i=0

b2i ⋅ xn−2id ⋅ ∣x∣2i.

We have

(H)(x) =⌊n/2⌋

∑i=0

b2i ⋅ [(xn−2id ) ⋅ ∣x∣2i + xn−2i

d ⋅ (∣x∣2i)]

=⌊n/2⌋

∑i=0

b2i ⋅ [(n − 2i)(n − 2i − 1) ⋅ xn−2i−2d ⋅ ∣x∣2i

+xn−2id ⋅ (∣x∣2i)] (6.20)


and

(∣x∣2i) =d−1

∑j=1

2i [∣x∣2(i−1) + 2x2j(i − 2)∣x∣2(i−2)]

= 2i(2i + d − 3)∣x∣2(i−1). (6.21)

Combining 6.20 and 6.21 yields

0 =⌊n/2⌋

∑i=0

[(n − 2i)(n − 2i − 1)b2i + 2(i + 1)(d + 2i − 1)b2i+2]xn−2i−2d ∣x∣2i.

Hence we obtain b2i = b2i−2 ⋅ (n−2i)(n−2i−1)(2i+2)(d+2i−1) for i = 1, . . . , ⌊n/2⌋.

Proposition 20. For every d ≥ 2 there exists a unique polynomial P nd (t) of degree n with

the following property: if HiN(d,n)i=1 is any orthonormal basis of Hnd and u, v are any two

unit vectors then ∑N(d,n)i=1 Hi(u)Hi(v) = N(d,n) ⋅ P n

d (⟨u, v⟩).

Proof. If d ≥ 3 then Hnd is trivially invariant under rotations and thus we may use Propo-

sition 2. Consider Q(⟨u, ed⟩) with Q(t) defined as in Proposition 2 then as a function in

the variable u ∈ Sd−1, it is invariant with respect to rotations leaving ed fixed. By Lemma

7, Q(⟨u, ed⟩) = C ⋅ Tn(u). This implies

Q(t) = Q (⟨(0, . . . ,√

1 − t2, t), ed⟩) = C ⋅⌊n/2⌋

∑i=0

a2i ⋅ tn−2i(1 − t2)i.

It is clear from Lemma 7 that an ≠ 0 and consequently Qn(t) is a polynomial of degree

n. Letting P nd (t) = Q(t) ⋅ 1/N(d,n) completes the proof for the case d ≥ 3. If d = 2, it is

well known from complex analysis that if H ∈ Hn2 there exists a complex number z1 such

that H(x) = Re(z1zn) with z = x1 + ix2 and z1 = a1 + b1i. Switching to polar coordinates

(0 ≤ τ < 2π), we have

H(u) =H(cos(τ), sin(τ)) = Re (z1einτ)

= a1 cos(nτ) + b1 sin(nτ).


If we let

H1(u) ∶= cos(narccos(u1)),H2(u) ∶= sin(narcsin(u2)) (6.22)

then we have

⟨H1,H2⟩ =1

2π ∫2π

0cos(nτ) sin(nτ)dτ = 0. (6.23)

In other words, H1,H2 is an orthonormal basis for Hn2 . Let u, v be two unit vectors

and set t = ⟨u, v⟩ then

H1(u)H1(v) +H2(u)H2(v) = cos(narccos(t)). (6.24)

To complete the proof, we let P n2 (t) = 1/2 ⋅ cos(narccos(t)).

Corollary 6. For any n ≥ 0, P nd (1) = 1 and ∣P n

d (t)∣ ≤ 1. In addition, if H ∈ Hnd then

∥H∥∞ ≤√N(d,n)∥H∥2.

Proof. Temporarily let N(d,n) = N then

P nd (1) = 1/N ⋅

N

∑i=1∫Sd−1

∣H(u)∣2dθ(u) = 1

and by the Cauchy-Schwartz inequality

∣P nd (t)∣ ≤

√P nd (1)

√P nd (1) = 1.

To prove the second part of the corollary, let H ∈ Hnd then

∣H(u)∣ = ∣N

∑i=1

⟨H,Hi⟩Hi(u)∣

≤

¿ÁÁÀ N

∑i=1

⟨H,Hi⟩2 ⋅

¿ÁÁÀ N

∑i=1

H2i (u)

= ∥H∥2 ⋅√P nd (1) ⋅N

=√N ⋅ ∥H∥2.


Theorem 10. (Funck-Hecke Theorem)Let f(t) be a real valued function defined on

[−1,1] such that ∫1

−1 f(t)(1 − t2)d−32 < ∞. If H a spherical harmonic of degree n and

u a fixed unit vector then the function f(⟨u, v⟩) is in L1(Sd−1) and

∫Sd−1

f(⟨u, v⟩)H(v)dθ(v) = αd,n(f)H(u)

with

αd,n(f) =γd−1

γd∫

1

−1f(t)P n

d (t)(1 − t2)d−32 dt.

Proof. If ∫1

−1 f(t)(1 − t2)d−32 <∞ then we have the following transformation formula

∫Sd−1

f(⟨u, v⟩)dθ(v) = γd−1

γd∫

1

−1f(t)(1 − t2) d−32 dt (6.25)

= ∫1

−1f(t)dµ∗(t) (6.26)

with the measure µ∗ implicitly defined by 6.25 and 6.26. In particular, if f(t) is in L1µ∗

then f(⟨u, v⟩) is in L1(Sd−1) for all unit vectors u. We will now show that the polynomials

P nd (t) are mutually orthogonal with respect to the inner product

⟨g1, g2⟩∗ ∶= ∫1

−1g1(t)g2(t)dµ∗(t).

Let m,n be two nonegative integers and let HiN(d,m)i=1 be an orthonormal basis for Hmd

and H ′jN(d,n)j=1 an orthonormal basis for Hnd then by Proposition 20 and the transforma-

tion formula 6.26

N(d,m)N(d,n) ⟨Pmd , P

nd ⟩∗ =

N(d,n)

∑j=1

N(d,m)

∑i=1

Hi(u)H ′j(u) ⟨Hi,H

′j⟩ . (6.27)


If m = n then by corollary 6

N2(d,n) ⟨P nd , P

nd ⟩∗ = N(d,n)P n

d (1) = N(d,n)

and if m ≠ n then by the Green-Gauss theorem

0 = ∫Bd

[Hm(x)Hn(x) −Hm(x)Hn(x)]dx

= γd∫Sd−1

[Hm(u) ⟨∇Hn(u), u⟩ − ⟨∇Hm(u), u⟩Hn(u)]dθ(u). (6.28)

A basic computation shows that if P (x) is a homogeneous polynomial of degree k then

⟨∇P (x), x⟩ = kP (x) ∀x.

As a consequence, the right-hand side of 6.28 simplifies to

γd(n −m) ⟨Hm,Hn⟩

and thus ⟨Hm,Hn⟩ = 0 whenever m ≠ n. In particular, the right-hand side of 6.27 vanishes

whenever m ≠ n. The orthogonality of the polynomials P nd (t) of differing degrees imply

that for every polynomial P (t) of degree N

P (t) =N

∑i=0

⟨P,P id⟩∗

N(d, i)P id(t).

If f(t) is in L1µ∗ then it is well known that there exists a sequence of polynomials QN(t)

of degree at most N such that QN(t) converges in L1µ∗ to f(t) as N tends to infinity.


This gives

limN→∞

⟨QN(⟨u, v⟩),H(v)⟩ = limN→∞

N

∑i=1

⟨QN , P id⟩∗

N(d, i)⟨P i

d(⟨u, v⟩),H(v)⟩

= H(u) ⋅ limN→∞

⟨QN , Pnd ⟩∗

= H(u) ⟨f,P nd ⟩∗ .

Noting that

limN→∞

⟨∣(QN − f)(⟨u, v⟩)∣, ∣H(v)∣⟩ ≤ limN→∞

∥H∥∞ ⋅ ∫1

−1∣f(t) −QN(t)∣dµ∗(t) = 0

implies

H(u) ⟨f,P nd ⟩∗ = lim

N→∞⟨QN(⟨u, v⟩),H(v)⟩ = ⟨f(⟨u, v⟩),H(v)⟩ .

We say that a function f defined on Sd−1 is zonal with pole p ∈ Sd−1 if it depends solely

on its distance from p. For any unit vector u, the distance between u and p depends

only ⟨u, p⟩. We thus deduce that a function f is zonal with pole p if and only if there

exists a function f1 defined on [−1,1] such that f(u) = f1(⟨u, p⟩). The next proposition

characterizes zonal spherical harmonics completely.

Proposition 21. A spherical harmonic H of degree n is a zonal harmonic with pole p

if and only if it is a constant multiple of P nd (⟨u, p⟩).

Proof. We know that there exists a continuous and bounded function h(t) such that

H(u) = h(⟨u, p⟩). We have H(u) = ∑N(d,n)i=1 ⟨H,Hi⟩Hi(u) and by Theorem 10

⟨H,Hi⟩ = ∫Sd−1

Hi(u)h(⟨u, p⟩)dθ(u) = αd,n(h)Hi(p).


This implies

H(u) = αd,n(h)N(d,n)

∑i=1

Hi(u)Hi(p) = αd,n(h) ⋅N(d,n) ⋅ P nd (⟨u, p⟩).

Corollary 7. If p ∈ Sd−1 and F is a continuous function on Sd−1 then for all H ∈ Hnd ,

∫Od F (ρ(p))H(ρ(u))dµ(ρ) = ⟨F,H⟩P nd (⟨u, p⟩).

Proof. If ρ ∈ Od then H(ρ(u)) ∈ Hnd and thus

∫OdF (ρ(p))H(ρ(u))dµ(ρ) =

Nd

∑i=1

Hi(u)∫Od

⟨H(ρ(u),Hi(u)⟩F (ρ(p))dµ(ρ).

This shows that ∫Od F (ρ(p))H(ρ(u))dµ(ρ) is in Hnd and since it is zonal with pole p it is

a constant multiple C of P nd (⟨u, p⟩). Recalling that P n

d (1) = 1 gives

C = ∫OdF (ρ(p))H(ρ(p))dµ(ρ)

= ∫Od∫Sd−1

F (ρ(p))H(ρ(p))dθ(p)dµ(ρ)

= ⟨F,H⟩ .

6.2.3 L2 Decay of Spherical Harmonics

We will now study the expected rate of decay of the L2 norm of spherical harmonics

under projections onto the random subspaces Snd (ρ) generated by the Haar measure. We

make the following remark: if H ′1, . . . ,H

′m is an orthonormal basis for Snd (I)∩Hdn then

H ′1 ρ−1, . . . ,H ′

m ρ−1 is an orthonormal subset of Snd (ρ) ∩Hnd and conversely if one

is given an orthonormal basis of Snd (ρ) ∩Hnd then composing the basis elements with ρ

yields an orthonormal subset of Snd (I)∩Hdn. In particular, the subspaces Snd (ρ)∩Hnd have


a common dimension which we denote by M(d,n).

Proposition 22. For any H ∈ Hnd , Eµ [∥Hρ∥22] =

M(d,n)N(d,n) ∥H∥2

2.

Proof. Temporarily let M(d,n) = M and N(d,n) = N . Suppose H ′1 . . . ,H

′M denotes

an orthonormal basis for Snd (I). If H ∈ Hnd then by the remark above

∫Od

∥Hρ∥22dµ(ρ) =

M

∑i=1∫Od

⟨H,H ′i ρ−1⟩2

dµ(ρ)

=M

∑i=1∫Od

⟨H ρ,H ′i⟩

2dµ(ρ). (6.29)

The right-hand side of equation 6.29 can be rewritten as

M

∑i=1∫Sd−1

∫Sd−1

∫OdH ′i(u)H ′

i(p)H(ρ(u))H(ρ(p))dµ(ρ)dθ(u)dθ(p)

which in turn equals (by using Corollary 7)

∥H∥22

M

∑i=1∫Sd−1

∫Sd−1

H ′i(u)H ′

i(p)P nd (⟨u, p⟩)dθ(u)dθ(p). (6.30)

By Proposition 20, we have

∫Sd−1

P nd (⟨u, p⟩)H ′

i(u)dθ(u) =H ′i(p) ⋅ 1/N

and consequently 6.30 equals

∥H∥22

M

∑i=1

∥H ′i∥2

2 ⋅ 1/N = M(d,n)N(d,n)

⋅ ∥H∥22. (6.31)

We will now compute M(d,n) and N(d,n) by using standard results from linear

algebra and combinatorics.


Proposition 23. M(d,n) = 0 if n is odd and M(d,n) =⎛⎜⎜⎝

d + n2 − 2

d − 2

⎞⎟⎟⎠

if n is even. In

addition, N(d,n) =⎛⎜⎜⎝

d + n − 2

d − 2

⎞⎟⎟⎠d+2n−2d+n−2 .

Proof. If n is odd and H ∈ Snd (I) then necessarily H(u) = H(−u) = −H(u) = 0 and thus

M(d,n) = 0. If n is even then H ∈ Snd (I) if and only the powers of H(x1, . . . , xd) in each

variable are even. If E(d,n) denotes the space of homogeneous polynomials of degree

n in d variables whose powers in each variable are even then the space Snd (I) ∩ Hnd is

precisely the restriction to Sd−1 of the kernel of the Laplacian restricted to E(d,n). Since

the image of E(d,n) under the Laplacian is precisely E(d,n − 2) then

M(d,n) = dim(E(d,n)) − dim(E(d,n − 2)). (6.32)

The dimension of E(d,n) is the same as the number of partitions of n/2 into at most

d parts and this in turn can be computed by reading off the coefficient of xn/2 in the

generating function

(1 + x + x2 +⋯)d = (1 − x)−d =∞∑k=0

⎛⎜⎜⎝

d + k − 1

d − 1

⎞⎟⎟⎠xk. (6.33)

Hence

Mnd =

⎛⎜⎜⎝

d + n2 − 1

d − 1

⎞⎟⎟⎠−⎛⎜⎜⎝

d + n2 − 2

d − 1

⎞⎟⎟⎠=⎛⎜⎜⎝

d + n2 − 2

d − 2

⎞⎟⎟⎠. (6.34)

By using the same arguments used to compute M(d,n) one gets

N(d,n) =⎛⎜⎜⎝

d + n − 1

d − 1

⎞⎟⎟⎠−⎛⎜⎜⎝

d + n − 3

d − 3

⎞⎟⎟⎠=⎛⎜⎜⎝

d + n − 2

d − 2

⎞⎟⎟⎠

d + 2n − 2

d + n − 2. (6.35)


6.2.4 L∞ Decay of Spherical Harmonics

Returning to the geometry, we recall that given any convex body K with R(K) ≤ c, we

have the geometric inequality


Bd)] ≤ 2cc3ε′ +

n

∑j=1

ETµ [∥Qρρρj∥∞] (6.36)

with n = ⌈ dε′ ⌉. By Corollary 6 and Jensen’s inequality (applied to the concave function

x1/2):

n

∑j=1

ETµ [∥Qρρρj∥∞] ≤

n

∑j=1

N(d, j)1/2ETµ [∥Qρρρj∥2] (6.37)

≤n

∑j=1

N(d, j)1/2 (ETµ [∥Qρρρj∥2

2])1/2. (6.38)

Recalling Proposition 22 yields

ETµ [∥Qρρρj∥2

2] =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

(M(d,j)N(d,j) )

T∥Qj∥2

2 if j is even

0 if j is odd

(6.39)

and by applying the Cauchy-Schwartz inequality to the right-hand side of 6.38 we obtain

n

∑j=1

ETµ [∥Qρρρ2j∥∞] ≤

⎛⎝

⌊n2⌋

∑j=1

N(d,2j)(M(d,2j)N(d,2j)

)T⎞⎠

1/2⎛⎝

⌊n2⌋

∑j=1

∥Q2j∥22

⎞⎠

1/2

. (6.40)

We now estimate⌊n2⌋

∑j=1


)T

(6.41)

by breaking it up into two pieces and estimating each piece separately.

Lemma 8. The following holds:

∑(d−2)≤2j≤n


)T

< (3

4)T (d−1)

ed(1 + 1/ε′)d−1. (6.42)


Proof. By Proposition 23

M(d,2j)N(d,2j)

= (d + 2j − 2

d + 4j − 2)

⎛⎜⎜⎝

d + j − 2

d − 2

⎞⎟⎟⎠

⎛⎜⎜⎝

d + 2j − 2

d − 2

⎞⎟⎟⎠

. (6.43)

We have⎛⎜⎜⎝

d + j − 2

d − 2

⎞⎟⎟⎠

⎛⎜⎜⎝

d + 2j − 2

d − 2

⎞⎟⎟⎠

=d−2

∏i=1

j + i2j + i

< ( d − 2 + jd − 2 + 2j

)d−2

. (6.44)

However, we also have that

( d − 2 + jd − 2 + 2j

)

is increasing in d for every fixed j and since by assumption 2j ≥ d − 2 then

d − 2 + jd − 2 + 2j

< 3

4

and

M(d,2j)N(d,2j)

< (3

4)d−1

(6.45)

for all 2j ≥ d − 2. In particular,

∑(d−2)≤2j≤n


)T

< (3

4)T (d−1)

∑0≤2j≤n

N(d,2j). (6.46)

We have the combinatorial identity

∑0≤2j≤n

N(d,2j) =⎛⎜⎜⎝

d + n − 1

d − 1

⎞⎟⎟⎠

(6.47)


and consequently we will need upper bounds on binomial coefficients to estimate the

right-hand side of 6.46. Recalling Stirling’s formula

n! = nne−n√

2πneθ(n)/12n (0 < θ(n) < 1) (6.48)

yields

⎛⎜⎜⎝

n

m

⎞⎟⎟⎠< (e ⋅ n

m)m

and in particular

⎛⎜⎜⎝

d + n − 1

d − 1

⎞⎟⎟⎠< (e ⋅ d + n − 1

d − 1)d−1

. (6.49)

Combining 6.46,6.47 and 6.49 and recalling that n = ⌈d/ε′⌉ finally gives

∑(d−2)≤2j≤n


)T

< (3

4)T (d−1)

ed−1 ( d

d − 1)d−1

(1 + 1/ε′)d−1

< (3

4)T (d−1)

ed(1 + 1/ε′)d−1.

We now estimate the other piece.

Lemma 9. The following holds:

∑2j<(d−2)


)T

≤e2 (1

2)T−2

1 − e2 (12)T−2

. (6.50)


Proof. We first use the combinatorial identity

⎛⎜⎜⎝

d + j − 2

d − 2

⎞⎟⎟⎠

⎛⎜⎜⎝

d + 2j − 2

d − 2

⎞⎟⎟⎠

=

⎛⎜⎜⎝

d + j − 2

j

⎞⎟⎟⎠

⎛⎜⎜⎝

d + 2j − 2

2j

⎞⎟⎟⎠

=j

∏i=1

j + id − 2 + j + i

< ( 2j

d − 2 + 2j)j

which gives (by recalling Proposition 23)


)T

<⎛⎜⎜⎝

d + 2j − 2

d − 2

⎞⎟⎟⎠( 2j

d − 2 + 2j)jT

. (6.51)

However, since 2jd−2+2j is increasing in j and 2j < d − 2 then

2j

d − 2 + 2j< 1

2. (6.52)

Recalling the inequality 6.49 for binomial coefficients and using 6.51 and 6.52 finally gives

∑2j<(d−2)


)T

< ∑2j<(d−2)

(e ⋅ d + 2j − 2

2j)

2j

( 2j

d + 2j − 2)jT

< ∑2j<(d−2)

⎡⎢⎢⎢⎣e(1

2)T−22 ⎤⎥⎥⎥⎦

2j

<∞∑j=1

⎡⎢⎢⎢⎣e(1

2)T−22 ⎤⎥⎥⎥⎦

2j

=e2 (1

2)T−2

1 − e2 (12)T−2

. (6.53)


6.2.5 Final Results

We see from 6.40 and Lemmas 8 and 9 that ETµ [δ (Mρρρ(K), b(K)2 Bd)] is bounded by:

⎛⎝

e2 (12)T−2

1 − e2 (12)T−2

+ (3

4)T (d−1)

ed(1 + 1/ε′)d−1⎞⎠

1/2⎛⎝

⌊n2⌋

∑j=1

∥Q2j∥22

⎞⎠

1/2

. (6.54)

However, we also have the basic inequality

⎛⎝

⌊n2⌋

∑j=1

∥Q2j∥22

⎞⎠

1/2

≤ ∥Pε′(K,u)∥2 ≤ ∥Pε′(K,u)∥∞ ≤ cc3ε′ + c. (6.55)

If we let

α(T, ε′, d) =⎛⎝

e2 (12)T−2

1 − e2 (12)T−2

+ (3

4)T (d−1)

ed(1 + 1/ε′)d−1⎞⎠

1/2

(cc3ε′ + c) (6.56)

then by using 6.54 and 6.55, we have


Bd)] < 2cc3ε′ + α(T, ε′, d) (6.57)

for all T ≥ 1. We let T (ε, d) denote the minimal number T such that

2cc3ε′ + α(ε′, T, d) ≤ ε (6.58)

for some ε′ > 0. If we put

ε′ = ε

4cc3

, ε′′ = ( 2ε

ε + 4c)

2

then it follows from 6.57 and 6.58 that if T is large enough to satisfy

e2 (12)T−2

1 − e2 (12)T−2

≤ ε′′/2, (6.59)

ed (1 + 1/ε′)d−1 (3

4)T (d−1)

≤ ε′′/2 (6.60)


then T (ε, d) ≤ T . Solving the inequalities 6.59 and 6.60 yields

T (ε, d) ≤ T1(ε) ∨ T2(ε, d) (6.61)

with

T1(ε) = 2 + 1

2[log(1 + ε/2

ε/2) − 2] ,

T2(ε, d) = 1

log(4/3)(d − 1)[d + (d − 1) log (1 + 4cc3

ε) + log(2/ε′′)] .

We immediately see that there exists a numerical constant c4 such that

T (ε, d) ≤ cc4 log(1/ε). (6.62)

We thus have:

Theorem 11. [9, p.1333]There exists a numerical constant c4 such that for all ε > 0

E⌈cc4d log(1/ε)⌉µ [δ (Mρρρ(K), b(K)

2Bd)] ≤ ε

for every convex body K satisfying R(K) ≤ c.

Corollary 8. [9, Theorem 1.3]There exists a numerical constant c4 such that for all

ε > 0 there exists at most ⌈cc4d log(1/ε)⌉ Minkowski symmetrizations that transform any

convex body K with R(K) ≤ c into a new convex body K ′ with the property

δ (K ′,b(K)

2Bd) ≤ ε.

Proposition 24 (Urysohn’s Inequality). b(K)2 ≥ 1 for every convex body K with volume

κd.


Proof. By Corollary 8, there exists a sequence of unit vectors ui∞i=1 such that

limn→∞

δ (Mu1,...,un(K), b(K)2

Bd) = 0. (6.63)

However, Minkowski symmetrization increases volume (since it contains Steiner sym-

metrization and the latter preserves volume) and thus by 6.63

κd ≤ limn→∞

Vd(Mu1,...,un(K)) = (b(K)/2)d κd. (6.64)

6.3 Steiner and Minkowski Symmetrization: Making

the Link.

6.3.1 Preliminaries

Notation and Preliminary Facts

If K is a convex body, we will denote by p(K,x) the closest point to x in K and we

will denote by u(K,x) the unit vector (x − p(K,x))d(x,K)−1 with d(x,K) the distance

between x and K. The support set of K with outer normal u will be denoted by

F (K,u) ∶= x ∈ ∂K ∶ ⟨x,u⟩ = h(K,u)

and the normal cone of x ∈ ∂K will be denoted by

N(K,x) = y ∶ p(K,y) = x.


Two polytopes P1, P2 are said to be strongly isomorphic if for every u ∈ Sd−1

dim (F (P1, u)) = dim (F (P2, u)) .

We say that a convex body is regular if its boundary does not contain any singularities

i.e., every point of the boundary is contained in a single support set. Lastly, we say that

a convex body is strictly convex if all of its support sets are single points.

Section Outline

Let K be a convex body with volume κd. It is clear that b(K)2 < R(K) unless K is a ball.

As explained in section 6.1.3 “Outline of the Strategy ”, we will need to quantify how small

b(K)2 is relative to R(K) in order to derive rates of convergence for Steiner symmetrization

from those for Minkowski symmetrizations derived in the previous section. The purpose

of this section is to give a self-contained derivation of the following result:

Proposition 25. [9, Theorem 6.1]Let ε > 0 and suppose K ⊂ B(0,1+ ε) is a convex body

with Vd(K) = κd then b(K)2 < 1 + (1 − 1/d2) ε. If ε < 1/d then b(K)

2 < 1 + (1 − 1/2d)ε.

6.3.2 Differential Geometry of Convex Bodies

We will now review some definitions and results from the differential geometry of convex

bodies. We must warn the reader that some proofs will be omitted. For a complete

treatment, the interested reader should consult chapters 2,4 and 5 of [15] from which the

presentation is largely based on.

Curvature Measures and Quermassintegrals

Let K be a d dimensional convex body then for every ξ > 0, we can define a measure

µξ(K, ⋅) defined on the collection B of Borel sets contained in the product space Sd−1×Rd


by letting

µξ(A) = Vd (x ∶ 0 < δ(x,K) ≤ ξ, (u(K,x), p(K,x)) ∈ A)

for every A ∈ B. The measures µξ(K, ⋅) have the nice property that they are weakly

continuous with respect to the Hausdorff metric on the space of convex bodies Kd. In

other words, if Kn is a sequence of convex bodies converging in Hausdorff metric to a

convex body K then for any open set U in Sd−1 ×Rd

lim infn→∞

µξ(Kn, U) ≤ µξ(K,U).

Let P be a fixed polytope and A ∈ B, we will now show that µξ(P,A) is a degree d − 1

polynomial in ξ. We denote by Fm the faces of P of dimension m = 0, . . . , d−1. If F ∈ Fm

then

µξ(P,A) =d−1

∑m=0

∑F ∈Fm

∫FVd−mt ⋅ u ∶ 0 ≤ t ≤ ξ, u ∈ N(P,F ), (u, y) ∈ AdVm(y)

=d−1

∑m=0

∑F ∈Fm

ξd−m

d −m⋅ ∫

FVd−m−1u ∶ (u, y) ∈ AdVm(y) (6.65)

=d−1

∑m=0

ξd−m⎛⎜⎜⎝

d − 1

m

⎞⎟⎟⎠

Θm(P,A) (6.66)

with the measures Θm(P,A) implicitly defined by 6.65 and 6.66. Setting ξ = 1, . . . , d in

6.66, we obtain

⎛⎜⎜⎜⎜⎜⎜⎝

Θ0(P,A)

⋮

Θd−1(P,A)

⎞⎟⎟⎟⎟⎟⎟⎠

= B−1

⎛⎜⎜⎜⎜⎜⎜⎝

µ1(P,A)

⋮

µd(p,A)

⎞⎟⎟⎟⎟⎟⎟⎠

, Bjm = jd−m

⎛⎜⎜⎝

d − 1

m

⎞⎟⎟⎠.


To extend the measures Θm to arbitrary convex bodies, we let

⎛⎜⎜⎜⎜⎜⎜⎝

Θ0(K, ⋅)

⋮

Θd−1(K, ⋅)

⎞⎟⎟⎟⎟⎟⎟⎠

= B−1

⎛⎜⎜⎜⎜⎜⎜⎝

µ1(K, ⋅)

⋮

µd(K, ⋅)

⎞⎟⎟⎟⎟⎟⎟⎠

.

From this definition and weak continuity of the measures µi(K, ⋅) i = 1, . . . , d, we deduce

that the measures Θm(K, ⋅) are also weakly continuous. Lastly, note that equation can be

extended to general convex bodies via approximation by polytopes and weak continuity

of the measures µξ(⋅,A) and Θm(⋅,A):

µξ(K,A) =d−1

∑m=0

ξd−m⎛⎜⎜⎝

d − 1

m

⎞⎟⎟⎠

Θm(K,A). (6.67)

We can now define the so called curvature measures by restricting the measures Θm

to specific sets in B. More precisely, for 0 ≤ m ≤ d − 1 and for any Borel subset β of ∂K

and Borel subset ω of Sd−1 we let

Cm(K,β) = Θm(K,Sd−1 × β), Sm(K,ω) = Θm(K,ω × ∂K).

We also let

σ(K,β) = ⋃x∈β

N(K,x) ∩ Sd−1, τ(K,ω) = ⋃u∈ω

F (K,u)

denote the spherical image map and reverse spherical image map respectively. For regular

and strictly convex bodies, one has

Cm(K,τ(K,ω)) = Sm(K,ω),Cm(K,β) = Sm(K,σ(K,β)). (6.68)

The Quermassintegrals Wm(K) for m = 1, . . . , d are defined in terms of the total


curvature measures for K:

Wm(K) = 1

d⋅Cd−m(K,∂K) = 1

d⋅ Sd−m(K,Sd−1). (6.69)

We also let W0 = Vd(K). With this definition, we have by 6.67

Vd(K + ξBd) =d

∑m=0

ξm⎛⎜⎜⎝

d

m

⎞⎟⎟⎠Wm(K). (6.70)

Mixed Volumes and Inequalities

We saw in the previous subsection that Vd(K + ξBd) is a polynomial in ξ. In this sub-

section, we will give an overview of how we can express Vd(K + ξBd) in a different way

using mixed volumes.

Let P be a d dimensional polytope with facets F1, . . . , Fm and corresponding outward

unit normal vectors u1, . . . , um. Suppose first that 0 is in the interior of the convex body

then Vd(P ) is the sum of the volume of m pyramids with base Fi and height h(P,ui) i.e.,

Vd(P ) = 1/d ⋅m

∑i=1

h(P,ui)Vd−1(Fi). (6.71)

It turns out that 6.71 holds even if 0 is not in the interior of the polytope. Indeed, one

can show the weighted average unit normal ∑mi=1 ui ⋅ Vd−1(Fi) = 0 and thus if x ∈K

Vd(P ) = Vd(P − x)

= 1

d⋅d

∑i=1

h(P − x,ui)Vd−1(Fi)

= 1

d⋅m

∑i=1

h(P,ui)Vd−1(Fi) − ⟨x,m

∑i=1

uiVd−1(Fi)⟩

= 1

d⋅m

∑i=1

h(P,ui)Vd−1(Fi). (6.72)


We now introduce some useful definitions and notation. We let (i, j) ∈ I if and only

if Fi ∩ Fj = Fij ≠ ∅. Fix a facet Fi and consider it as a d − 1 polytope lying in its d − 1

dimensional ambient space. The facets of Fi are Fij with (i, j) ∈ I and have corresponding

unit normals vij. Lastly, we let hi = h(Fi, ui) = h(P,ui) and hij = h(Fi, vij). Clearly hi is

orthogonal to vij and ⟨uj, vij⟩ > 0. A straightforward computation yields for any v ∈ Fi:

⟨v, vij⟩ =⟨v, uj⟩ − ⟨ui, uj⟩ ⋅ ⟨v, ui⟩

⟨vij, uj⟩

=⟨v, uj⟩ − ⟨ui, uj⟩ ⋅ hi

⟨vij, uj⟩. (6.73)

If we maximize the left hand side of 6.73 with respect to v, we obtain

hij =hj − ⟨ui, uj⟩ ⋅ hi

⟨vij, uj⟩. (6.74)

Using 6.72 and equation 6.74, it is a straightforward inductive argument to show that

for any d dimensional polytope P , Vd(P ) can be written as a degree d homogeneous

polynomial in the variables h(P,ui) whose coefficients are invariant under permutation

of indices and equal to the coefficients of the polynomial Vd(P ′) in the variables h(P ′, ui)

whenever P ′ is strongly isomorphic to P . If P1, . . . , Pn are strongly isomorphic d dimen-

sional polytopes then λ1K1 +⋯ + λnKn is strongly isomorphic to Pi for all 1 ≤ i ≤ n and

thus

Vd(λ1P1 +⋯ + λnKn) =d

∑i1,...,im=1

ai1,...,idhi1⋯hid

=d

∑i1,...,im=1

ai1,...,id

d

∏j=1

(m

∑r=1

λrhrij)

=d

∑i1,...,im=1

ai1,...,id (n

∑r1,...,rd=1

λr1⋯λrd ⋅ hr1i1⋯hrdid)

=n

∑r1,...,rd=1

λr1⋯λrd (d

∑i1,...,im=1

ai1,...,id ⋅ hr1i1⋯hrdid) .


If we define

V (P1, . . . , Pd) =d

∑i1,...,im=1

ai1,...,id ⋅ h1i1⋯h

did

(6.75)

as the mixed volume of the strongly isomorphic polytopes P1, . . . , Pm then we obtain the

formula

Vd (λ1P1 +⋯ + λmPm) =m

∑r1,...,rd=1

λr1⋯λrdV (Pr1 , . . . , Prd). (6.76)

In analogy with the previous subsection, one will generalise the notion of a mixed

volume to general convex bodies by first writing the mixed volume for strongly isomorphic

polytopes P1, . . . , Pd as a linear expression in the variables Vd(Pi1 +⋯+Pij) with 1 ≤ j ≤ d

and then use an approximation result. In fact, one can check that

V (P1, . . . , Pd) =1

d!

d

∑k=1

(−1)k+d ⋅ ∑i1<⋯<ik

Vd(Pi1 +⋯ + Pid). (6.77)

If K1, . . . ,Kd are convex bodies, we let

V (K1, . . . ,Kd) =1

d!

d

∑k=1

(−1)k+d ⋅ ∑i1<⋯<ik

V (Ki1 +⋯ +Kid) (6.78)

define the mixed volume of K1, . . . ,Kd. Note that 6.78 shows mixed volumes are always

nonnegative. According to [15, p.102], for every sequence K1, . . . ,Kd of convex bodies,

we can choose a sequence of d dimensional polytopes P ri ∞i=1 such that P r

i are strongly

isomorphic for all i ≥ 1 and P ri converges to Kr for all 1 ≤ r ≤ d. From this approximation

result, one easily extends 6.76 to general sequences K1, . . . ,Kd:

V (λ1K1 +⋯ + λmKm) =m

∑r1,...,rd=1

λr1⋯λrdV (Kr1 , . . . ,Krd). (6.79)

We can simplify 6.79 by introducing some useful notation. We let

V (K1[j1], . . . ,Km[jm])


denote the mixed volume of the sequence of d convex bodies with K1 appearing j1 times,

K2 appearing j2 times and so forth and let

⎛⎜⎜⎝

d

j1, . . . , jm

⎞⎟⎟⎠= n!

j1!⋯jm!

denote the number of ways of choosing d elements from m different classes of d elements

with j1 chosen from the first class, j2 from the second and so forth. With this notation,

the right-hand side of 6.79 simplifies to

∑j1+⋯+jm=d

λj11 ⋯λjmm⎛⎜⎜⎝

d

j1, . . . , jm

⎞⎟⎟⎠V (K1[j1], . . . ,Km[jm]). (6.80)

To conclude the discussion of mixed volumes, we present a famous inequality due to

Minkowski that will be used in the final section of this chapter.

Mixed Area Measures

Let P1, . . . , Pd be a sequence of strongly isomorphic d dimensional polytopes then using

the notation of the previous subsection, we know from equation 6.76 that

Sd−1(λ1P1 +⋯ + λmPm, ω) = ∑ui∈ω

Vd−1(F (λ1P1 +⋯ + λmPm, ui))

= ∑ui∈ω

m

∑j1,...,jd−1=1

λj1⋯λjd−1v(Fj1i , . . . , F

jd−1i )

=m

∑i1,...,id−1=1

λi1⋯λid−1 ∑ui∈ω

v(F j1i , . . . , F

jd−1i ).

with v(⋅, . . . , ⋅) the mixed volume in d− 1 dimensions. Hence if P1, . . . , Pd−1 is a sequence

of strongly isomorphic d dimesonal polytopes and we let

S(P1, . . . , Pd−1, ⋅) =∑ui∈⋅

v(F 1i , . . . , F

d−1i ) (6.81)


denote the mixed area measure for P1, . . . , Pd−1 then we have the formula

Sd−1(λ1P1 +⋯ + λmPm, ⋅) =m

∑j1,...,jd−1=1

λi1⋯λid−1S(Pj1 , . . . , Pjd−1 , ⋅). (6.82)

One can show using induction on the dimension d that 6.72 extends to mixed volumes

in the sense that

v(P1, . . . , Pd) = 1

d

n

∑i=1

h1i v(F 2

i , . . . , Fdi )

= 1

d ∫Sd−1h(P1, u)dS(P2, . . . , Pd, u). (6.83)

We wish to extend area measures to general sequences K1, . . . ,Kd−1. To do so, we

copy the strategy used to extend mixed volumes to general sequences of convex bodies.

Firstly, one checks that

S(P1, . . . , Pd−1, ⋅) =1

(d − 1)!

d−1

∑k=1

(−1)k+d−1 ⋅ ∑i1<⋯<ik

Sd−1(Pi1 +⋯ + Pid−1 , ⋅) (6.84)

and secondly we define

S(K1, . . . ,Kd−1, ⋅) =1

(d − 1)!

d−1

∑k=1

(−1)k+d−1 ⋅ ∑i1<⋯<ik

Sd−1(Ki1 +⋯ +Kid−1 , ⋅) (6.85)

to be the mixed area measure of the convex bodies K1, . . . ,Kd−1.

We deduce from 6.85 that area measures are positive measures. Furthermore, from

the weak continuity of curvature measures we deduce that the mixed area measures are

weakly continuous. Approximating by strongly isomorphic polytopes and appealing to

the weak continuity of both Sd−1 and of S, we can extend equation 6.82:

Sd−1(λ1K1 +⋯ + λmKm, ⋅) =m

∑j1,...,jd−1=1

λi1⋯λid−1S(Kj1 , . . . ,Kjd−1 , ⋅). (6.86)


As for the mixed volumes, the right-hand side of equation 6.86 can be simplified to

∑j1+⋯+jm=d−1

λj11 ⋯λjmm⎛⎜⎜⎝

d − 1

j1, . . . , jm

⎞⎟⎟⎠S(K1[j1], . . . ,Km[jm], ⋅). (6.87)

Lastly, we wish to extend 6.83 to general sequences of convex bodies K1, . . . ,Kd. To

achieve this extension, we can choose a sequence of polytopes P ri ∞i=1 such that P r

i are

strongly isomorphic for all i ≥ 1 and P ri converges to Kr for all 1 ≤ r ≤ d. This implies

that h(P 1i , u) converges uniformly to h(K1, u) as i tends to infinity and by the weak

continuity of area measures one has

limi→∞∫Sd−1

f(u)dS(P 2i , . . . , P

di , u) = ∫

Sd−1f(u)dS(K2, . . . ,Kd, u) (6.88)

for every continuous function f(u). This implies

d ⋅ V (K1, . . . ,Kd) = limi→∞∫Sd−1

h(P 1i , u)dS(P 2

i , . . . , Pdi , u)

= limi→∞∫Sd−1

h(P 1i , u) − h(K1, u)dS(P 2

i , . . . , Pdi , u)

+ limi→∞∫Sd−1

h(K1, u)dS(P 2i , . . . , P

di , u)

= limi→∞∫Sd−1

h(P 1i , u) − h(K1, u)dS(P 2

i , . . . , Pdi , u)

+ ∫Sd−1

h(K1, u)dS(K2, . . . ,Kd).

Note that S(P 2i , . . . , P

di , S

d−1) is uniformly bounded since it converges to S(K2, . . . ,Kd, Sd−1)

and since h(P 1i , u) converges uniformly to h(K1, u) then

limi→∞∫Sd−1

h(P 1i , u) − h(K1, u)dS(P 2

i , . . . , Pdi , u) = 0

and finally

V (K1, . . . ,Kd) =1

d⋅ ∫

Sd−1h(K1, u)dS(K2, . . . ,Kd). (6.89)


Integral Representations of Quermassintegrals

In this subsection, we will consider the quermassintegrals of convex bodies with nice

boundaries. Suppose K has a Ck boundary then we can consider the bijective map

v ∶ ∂K → Sd−1 which maps x to its unique outward unit normal. If the inverse exists v−1

and is C1 then we say that K is a Ck+ body. Let K be a C2

+ body and temporarily let

h(K,u) = h(u). If N(y) is any C2 local parametrization of Sd−1 then v−1(x/∣x∣) = ∇h(x)

for all x ∈ Rd (see [15, p.40]) implies

∇(v−1 N) = ∇2h ∇N (6.90)

with ∇2h the Hessian of h. Note that the Hessian exists since we are assuming that v−1

is C1. From ∇h(x) = v−1(x/∣x∣) we deduce that

∇2h(u)(u) = 0 ∀u ∈ Sd−1. (6.91)

We let Tu denote the tangent space of u spanned by Ni(y)d−1i=1 with Ni = ∂iN . Note

that X = v−1 N is a local C2 parametrization of ∂K and

⟨∂iX(y), u⟩ = ⟨∇2h(u)(Ni(y)), u⟩ = ⟨Ni(y),∇2h(u)(u)⟩ = 0. (6.92)

This implies that

Xi(y) = ∂iX(y) =d−1

∑j=1

(∇v−1(u))jiNj(y) (i = 1, . . . , d − 1) (6.93)

for some real valued matrix (∇v−1(u))ji . If η = ηji (y) denotes the matrix

⟨Ni(y),Nj(y)⟩ acting as a linear transformation of Tu then

Xi =d−1

∑j=1

⟨Xi,Nk⟩ η−1 Nj


and consequently

∇v−1 = b η−1 (6.94)

with b the matrix

bji = ⟨Xi,Nj⟩ = ⟨∇2h Ni,Nj⟩ = ⟨Ni,∇2h Nj⟩ = bij. (6.95)

In particular, we see from 6.94 and 6.95 that ∇v−1 is symmetric. It is well known [6,

pp.102-103] that for any integrable f ∶ ∂K → R and integrable g ∶ Sd−1 → R

∫X(M)

f(x)dVd−1(x) = ∫Mf(X(y))

√det(γ(y))dVd−1(y) (6.96)

∫N(M)

g(u)dVd−1(u) = ∫Mg(N(y))

√det(η(y))dVd−1(y) (6.97)

with γ(y) the matrix γ(y)ji = ⟨Xi(y),Xj(y)⟩. We compute

⟨Xi(y),Xj(y)⟩ =d−1

∑k=1

d−1

∑r=1

⟨Nk,Nr⟩∇v−1(u)ki ⋅ ∇v−1(u)rj

= (∇v−1(u) η(y) ∇v−1(u))ji .

In particular, we have

det(γ(y)) = det(∇v−1(u))2 det(η(y)). (6.98)

Combining 6.96, 6.97 with 6.98 yields

∫X(M)

f(x)dVd−1(x) = ∫N(M)

f(v−1(u))det(∇v−1(u))dVd−1(u) (6.99)

∫N(M)

g(u)dVd−1(u) = ∫X(M)

g(v(x))det(∇v(x))dVd−1(x). (6.100)


If A denotes a symmetric d × d matrix with eigenvalues λ1, . . . , λd we let

sq(A) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

∑i1<⋯<iq λi1⋯λiq if 1 ≤ q ≤ d,

1 if q = 0

denote the qth elementary symmetric function of A. A simple calculation shows that

sj(A) =sd−jsd

(A−1) . (6.101)

We consider the Hessian ∇2h(u) with respect to the basis

N1(y), . . . ,Nd−1(y), u

for Rd. By 6.90 and 6.93, we see that the restriction of ∇2h(u) to Tu is ∇v−1(u) and

since u belongs to the kernel of ∇2h(u) then clearly

sj (∇v−1(u)) = sj (∇2h(u)) (j = 1, . . . , d − 1). (6.102)

As a result, 6.99 and 6.100 transform to

∫X(M)

f(x)dVd−1(x) = ∫N(M)

f(∇h(u))sd−1(∇2h(u))dVd−1(u) (6.103)

∫N(M)

g(u)dVd−1(u) = ∫X(M)

g(v(x))s−1d−1(∇2h(v(x))dVd−1(x). (6.104)

LetX be a C2 local parametrization of ∂K and letN = vX define a local parametrization

of Sd−1. We will compute µξ(K,Sd−1 ×X(M)). We have

µξ(K,Sd−1 ×X(M)) = V (X(y) +N(y) ∶ y ∈M,0 < t ≤ ξ) . (6.105)


In other words, we want to find the volume of the image of M × (0, ξ) under the map

φ(y, t) =X(y) + tN(y). The Jacobian Jφ evaluated at (y, t) is

det ((X1(y) + tN1(y), . . . ,Xd−1(y) + tNd−1(y),N(y)))

and since Xi(y) = ∇2h(u)(Ni(y)), we obtain

Jφ(y, t) = det(∇2h + tI) ⋅ det((N1, . . . ,Nd−1,N)). (6.106)

Since N(y) is orthogonal to N1(y), . . . ,Nd−1(y), it is well known

[1, pp.193-194] that

det((N1, . . . ,Nd−1,N)) =√

det(η) (6.107)

and by computing in an orthonormal basis we have

det((N(y)) + tI) =d−1

∑m=0

td−1−msm(∇2h(N(y))). (6.108)

Using 6.106,6.107 and 6.108, we deduce that µξ(K,Sd−1 ×X(M)) equals

∫M∫

ξ

0

d−1

∑m=0

td−m−1sm(∇2h(N(y)))√

det(eij(y))dtdy

which in turn equals

d−1

∑m=0

ξd−m

d −m ∫Msm(∇2h(N(y)))

√det(eij(y))dy (6.109)

By 6.97, 6.109 transforms to

d−1

∑m=0

ξd−m

d −m ∫σ(X(M))

sm(∇2h(u))dVd−1(u) (6.110)


and by comparing with 6.67 and using 6.68,6.97 and 6.101 we get

⎛⎜⎜⎝

d − 1

m

⎞⎟⎟⎠Cm(K,X(M)) = 1

d −m ∫σ(X(M))

sm(∇2h(u))dVd−1(u) (6.111)

and

⎛⎜⎜⎝

d − 1

m

⎞⎟⎟⎠Sm(K,σ(X(M))) = 1

d −m ∫X(M)

sd−1−m(Dv(x))dVd−1(x). (6.112)

The open neighborhoods of ∂K and Sd−1 generate the Borel sigma algebras of ∂K and

Sd−1 respectively and thus by the monotone class theorem, 6.111 and 6.112 extend to

arbitrary Borel sets β ⊂ ∂K and ω ⊂ Sd−1. In particular

i ⋅⎛⎜⎜⎝

d

i

⎞⎟⎟⎠⋅Wi(K) = ∫

Sd−1sd−i(∇2h(u))dVd−1(u) (6.113)

= ∫∂Ksi−1(Dv(x))dVd−1(x). (6.114)

We will now derive another useful integral representation for the Quermassintegrals

by using mixed volumes and mixed areas. Recalling 6.80 and 6.87,

V (K + ξBd) =d

∑m=1

ξd−m⎛⎜⎜⎝

d

m

⎞⎟⎟⎠V (K[m],Bd[d −m]), (6.115)

Sd−1(K + ξBd, ⋅) =d−1

∑m=0

ξd−1−m⎛⎜⎜⎝

d − 1

m

⎞⎟⎟⎠S(K[m],Bd[d − 1 −m], ⋅) (6.116)

and comparing with 6.70 and 6.67 (in dimensions d − 1), we obtain

Wi(K) = V (K[d − i],Bd[i]) (6.117)


and

S(K[m],Bd[d − 1 −m], ⋅) =⎛⎜⎜⎝

d − 1

m

⎞⎟⎟⎠

−1

1

d −m ∫⋅sm(∇2h(u))dVd−1(u). (6.118)

Combining 6.117, 6.118 as well as 6.89 yields

Wi(K) = V (K[d − i],Bd[i])

= 1

d ∫Sd−1h(K,u)dS(K[d − i − 1],Bd[i], u)

= 1

d(i + 1)

⎛⎜⎜⎝

d − 1

d − 1 − i

⎞⎟⎟⎠

−1

∫Sd−1

h(K,u)sd−i−1(∇2h(u))dVd−1(u)

= 1

i + 1

⎛⎜⎜⎝

d

i + 1

⎞⎟⎟⎠

−1

∫Sd−1

h(K,u)sd−i−1(∇2h(u))dVd−1(u). (6.119)

6.3.3 Inequalities between Quermassintegrals

We are now ready to give a complete presentation of a paper of Heil and Bukowski [2]

that yields inequalities between the quermassintegrals Wi. These inequalities serve as

the main tool to prove Proposition 25. We begin with a simple lemma:

Lemma 10. Suppose F (x) and f(x) are C1 and homogeneous of degree p and q respec-

tively then

∫Sd−1

f(u) ⋅ divF (u)dθ(u)

equals

(p + q + d − 1)∫Sd−1

[⟨f(u)F (u), u⟩ − ⟨∇f(u), F (u)⟩]dθ(u).

Proof. By the divergence theorem, we have

γd∫Sd−1

⟨f(u)F (u), u⟩dθ(u) = ∫B(0,1)

[f(x)divF (x) + ⟨∇f(x), F (x)⟩]dx. (6.120)


By switching to polar coordinates and by using the homogeneity of f and F 6.120 equals

γd∫Sd−1

∫1

0[rpf(u)rq−1divF (u) + ⟨rp−1∇f(u), rqF (u)⟩]rd−1drdθ(u). (6.121)

A computation shows that 6.121 equals

γd ⋅ (p + q + d − 1) ⋅ ∫Sd−1

f(u)divF (u) + ⟨∇f(u), F (u)⟩dθ(u).

If A denotes a symmetric d × d matrix with eigenvalues λ1, . . . , λd, we let

Tq(A) =q

∑k=0

(−1)k ⋅ sq−k(A) ⋅Ak (6.122)

denote the qth Newtonian transformation of A. In the following lemma, the symbol

sgn

⎛⎜⎜⎝

i1 ⋯ iq

j1 ⋯ jq

⎞⎟⎟⎠

will denote the sign of the permutation σ(ir) = jr and will be 0 other-

wise.

Lemma 11. The following holds [14, pp.373-375]:

i) sq (A) = 1q! ∑ sgn

⎛⎜⎜⎝

i1 ⋯ iq

j1 ⋯ jq

⎞⎟⎟⎠Aj1i1⋯A

jqiq

ii) Tq(A)ji = 1q! ∑ sgn

⎛⎜⎜⎝

i1 ⋯ iq i

j1 ⋯ jq j

⎞⎟⎟⎠Aj1i1⋯A

jqiq

iii) Tr(Tq(A) ⋅A) = (q + 1)Sq+1(A)

iv) Tq(A) = sqI − Tq−1(A) A

v) Tr (Tq(A)) = (d − q) ⋅ sq(A)

vi) If f is C3 then ∑di=1 ∂iTq(∇2f)ji = 0,∀j = 1, . . . , d.


vii) If f is C3 then q ⋅ sq(∇2f) = div (Tq−1(∇2f)(∇f)) .

Proof. i) Follows from the well-known representation of sq(A) in terms of principal mi-

nors. ii) follows by first computing in an orthonormal basis of eigenvectors and noting

that both sides of ii) transform similarily. To prove iii), we have

Tr(Tq(A) A) =d

∑i=1

d

∑j=1

Tq(A)ji ⋅Aij

= 1

q!

d

∑i=1

d

∑j=1

∑ δ

⎛⎜⎜⎝

i1 ⋯ iq i

j1 ⋯ jq j

⎞⎟⎟⎠Aj1i1⋯A

jqiqAji

= 1

q!⋅ (q + 1)! ⋅ sq+1(A)

= (q + 1) ⋅ sq+1(A).

iv) is a trivial computation and v) follows from iii) and iv). To prove vi), we compute

d

∑i=1

∂iTq(∇2f)ji = 1

q!

d

∑i=1

∑ δ

⎛⎜⎜⎝

i1 ⋯ iq i

j1 ⋯ jq j

⎞⎟⎟⎠

q

∑k=1

fi1j1⋯fikjki⋯fiqjq

= 1

q!

d

∑i=1

q

∑k=1

∑ δ

⎛⎜⎜⎝

i1 ⋯ iq i

j1 ⋯ jq j

⎞⎟⎟⎠fi1j1⋯fikjki⋯fiqjq

= 1

(q − 1)!

d

∑i=1

∑ δ

⎛⎜⎜⎝

i1 ⋯ iq i

j1 ⋯ jq j

⎞⎟⎟⎠fi1j1⋯fiqjqi

= −1

(q − 1)!

d

∑i=1

∑ δ

⎛⎜⎜⎝

i1 ⋯ i iq

j1 ⋯ jq j

⎞⎟⎟⎠fi1j1⋯fijqiq

= −d

∑i=1

∂iTq(∇2f)ji

= 0.


To prove vii), we have by vi) and iii)

div (Tq−1(∇2f)(∇f)) =d

∑i=1

d

∑j=1

∂i (Tq−1(∇2(f))ji∂jf)

=d

∑i=1

d

∑j=1

[∂i (Tq−1(∇2f)ji)∂jf + Tq−1(∇2f)ji ⋅ ∂i∂jf]

=d

∑i=1

d

∑j=1

Tq−1(∇2f)ji ⋅ ∂i∂jf

= Tr (Tq−1(∇2f) ∇2f)

= q ⋅ sq(∇2f).

Proposition 26. [2, Theorem 1]Let K be a convex body whose support function is C3

then

i

⎛⎜⎜⎝

d

i

⎞⎟⎟⎠Wi−2(K) = γd ⋅

1

i − 1⋅ ∫

Sd−1[sd−i(∇2h(u))(ih2(u) − ∣∇h(u)∣2)

+ ⟨Td−i−1(∇2h(u)) (∇h(u)) ,∇h(u)⟩ ]dθ(u)

Proof. We have by 6.113

(i − 1)⎛⎜⎜⎝

d

i − 1

⎞⎟⎟⎠Wi−2 = γd∫

Sd−1h(u)sd−i+1(u)dθ(u) (6.123)

and by part vii of Lemma 11 the right-hand side of 6.123 equals

1

d − i + 1⋅ γd∫

Sd−1h(u) ⋅ div (Td−i(∇2h(u))(∇h(u)))dθ(u). (6.124)

In order to use lemma 10, we need to consider the homogeneity of h and Td−i(∇2h(u))(∇h(u)).

Clearly h has homogeneity 1 and as a consequence ∇(h),∇2h and sq(∇2h) have homo-

geneity 0,−1 and −q respectively. This implies that Td−i(∇2h(u))(∇h(u)) has homogene-


ity i − d and thus 6.124 transforms to

γd ⋅1

d − i + 1⋅ ∫

Sd−1[i ⟨h(u)Td−i(∇2h(u))(∇h(u)), u⟩

− ⟨∇h(u), Td−i(∇2h)(∇h(u))⟩ ]dθ(u). (6.125)

Recalling 6.91, we obtain the relations

Tq(∇2h(u))(u) = u ⋅ sq(∇2h(u)),

⟨h(u)Td−i(∇2h(u))(∇h(u)), u⟩ = h(u) ⟨∇h(u), Td−i(∇2h(u))(u)⟩

= h2(u) ⋅ sd−i(∇2h(u)).

Using the above relations as well as part iv) of Lemma 11, 6.125 equals

γd ⋅1

d − i + 1⋅ ∫

Sd−1[i ⋅H2(u) ⋅ sd−i(∇2h(u)) − ⟨∇h(u),∇h(u) ⋅

sd−i(∇2h(u)) − Td−i−1(∇2h) ∇2h(u)(∇h(u))⟩]dθ(u)

which in turn equals

γd ⋅1

d − i + 1⋅ ∫

Sd−1[sd−i(∇2h(u))(i ⋅H2(u) − ∣∇h(u)∣2)

+⟨∇h(u), Td−i−1(∇2h) ∇2h(u)(∇h(u))⟩]dθ(u).

To complete the proof, note that

i

⎛⎜⎜⎝

d

i

⎞⎟⎟⎠= (d − i + 1)

⎛⎜⎜⎝

d

i − 1

⎞⎟⎟⎠.

It will now be shown how one can use these special integral representation to obtain in-


equalities between the Wi. Before doing so, we consider the notion of a concave sequence.

We say that a sequence asms=1 is concave if the value of every term in the sequence is

larger than the average of the values preceeding and proceeding it i.e., ai−1+ai+12 ≤ ai for

every i = 2, . . . ,m. From this definition, one easily deduces that for every 1 ≤ i < j < k ≤m

aj − aij − i

≥ak − ajk − j

⇔ ai(j − k) + aj(k − i) + ak(i − j) ≥ 0. (6.126)

Proposition 27. [2, Theorem 2]For every convex body K and for every 0 ≤ i < j < k ≤

d, the sequence (s + 1)Rs(K)Ws(K)ds=0 is concave and Ri(K)Wi(K)(i + 1)(j − k) +

Rj(K)Wj(K)(j + 1)(k − i) +Rk(K)(k + 1)Wk(K)(i − j) ≥ 0.

Proof. Since every convex body K can be approximated by C∞ bodies one may assume

that K is C∞ and by scaling one may set R = 1. It suffices to show that

i

⎛⎜⎜⎝

d

i

⎞⎟⎟⎠[(i + 1)Wi − 2(i)Wi−1 + (i − 1)Wi−2] (6.127)

is nonnegative. By 6.114,6.119 and Proposition 26, 6.127 equals

γd ⋅ ∫Sd−1

[sd−i(∇2h(u)) ((i + 1) − 2ih(u) + ih2(u) − ∣∇h(u)∣2)

+⟨∇h(u), Td−i−1(∇2h) ∇2h(u)(∇h(u))⟩]dθ(u).

Since ∇(h)(u) belongs to the boundary of K and K is contained in the unit ball then

∣∇(h)(u)∣2 ≤ 1 and in particular

(i + 1) − 2ih(u) + ih2(u) − ∣∇h(u)∣2 = i(h(u) − 1)2 + (1 − ∣∇h(u)∣2) ≥ 0.

It follows Lemma 11 that there exists an orthonormal basis whose elements are eigenvec-

tors (with nonnegative eigenvalues) for both Td−i−1(∇2h(u)) and ∇2h(u). This implies


that Td−i−1(∇2h) ∇2h(u) is positive definite and in particular

⟨∇h(u), Td−i−1(∇2h) ∇2h(u)(∇h(u))⟩ ≥ 0. (6.128)

The second statement in the proposition follows from 6.126.

6.3.4 The Key Estimates

Proof of Proposition 25. By considering i = 0, j = d − 1, k = d in Proposition 27, we have

the following inequality

W0 − d2 ⋅ (1 + ε)d−1 ⋅Wd−1 + (d2 − 1)(1 + ε)d ⋅Wd ≥ 0. (6.129)

Since W0(K) = κd,Wd−1(K) = κd ⋅ b(K)2 and Wd(K) = κd then inequality 6.129 yields

b(K)2

≤ 1/(d2 ⋅ (1 + ε)d−1) + (1 − 1/d2) ⋅ (1 + ε) (6.130)

< 1/d2 + (1 − 1/d2) ⋅ (1 + ε)

= 1 + ε ⋅ (1 − 1/d2).

If ε < 1/d then the inequality can be improved by estimating (1 + ε)1−d via its Taylor

expansion and using Leibniz’s theorem. More precisely, we have for ε < 1/d

(1 + ε)1−d = 1 +∞∑k=1

(−1)k(d − 1)⋯(d − 1 + k) ⋅ 1

k!⋅ εk (6.131)

=∞∑k=0

(−1)k ⋅ ak ⋅ εk (6.132)

with ak∞k=1 implicitly defined by 6.131 and 6.132. We note that ak ⋅εk∞k=0 is a decreasing

sequence if and only if ε < 1/d ≤ 1+kd+k for all k. Hence for ε < 1/d, (1 + ε)1−d < 1 − (d − 1) ⋅


ε + d(d−1)2 ⋅ ε2 and by 6.130

b(K)2

< 1

d2[1 − (d − 1)ε + d(d − 1)

2ε2] + (1 + ε)(1 − 1/d2)

= 1 + ε − εd+ d(d − 1) ⋅ ε2

2d2

< 1 + ε − εd+ ε

2

2

< 1 + ε − ε

2d.

6.4 Steiner Symmetrizations

6.4.1 Circumradius

In order to use the results from section 11, we first use Theorem 9 and apply 3d Steiner

symmetrizations to transform any convex body K into an isomorphic ball satisfying the

property

c1Bd ⊂K ′ ⊂ c2B

d (6.133)

for some numerical constants c1 and c2. In other words, by initially applying 3d Steiner

symmetrizations, we may assume from now on that K satisfies property 6.133.

We recall that Su(K) ⊂Mu(K) for all u ∈ Sd−1 and thus if we let

Sρ(K) = Sρ1,...,ρd(K) (6.134)

for all ρ = (ρ1, . . . , ρd) ∈ O and

Sρρρ(K) =Ti=1Sρi(K) (6.135)


for all ρρρ = (ρ1, . . . , ρT ) ∈ O ×⋯ ×O (T times) then

ETµ [R(Sρρρ(K))] ≤ ETµ [R(Mρρρ(K))] (6.136)

for all T ≥ 1. By Theorem 11 and 6.136, we have

ETµ [R(Sρρρ(K))] ≤ ε + b(K)2

(6.137)

for T ≥ c2c4d log(1/ε). However, by Proposition 25, we also have

b(K)2

<

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1 + (1 − 1/d2)(R(K) − 1) if R(K) − 1 ≥ 1/d

1 + (1 − 1/2d)(R(K) − 1) if R(K) − 1 < 1/d(6.138)

and thus by 6.137 and 6.138 if

T ≥

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

c2c4d log ( 2d2

(R(K)−1)) , for R(K) ≥ 1 + 1d

c2c4d log ( 4d(R(K)−1)) , for R(K) < 1 + 1

d

(6.139)

then

ETNµ [R(Sρρρ(K))] <

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

(1 − 12d2

)N (R(K) − 1) + 1 for R(K) ≥ 1 + 1d

(1 − 14d)N (R(K) − 1) + 1 for R(K) < 1 + 1

d

(6.140)

for all N ∈ N. We easily deduce from 6.139 and 6.140 that if R(K) ≥ 1 + ε ≥ 1 + 1/d then

ETNµ [R(Sρρρ(K))] ≤ 1 + ε (6.141)

whenever

N ≥log ( c2−1

ε)

log ( 2d2

2d2−1)≥

log (R(K)−1ε )

log ( 2d2

2d2−1)

(6.142)


and

T ≥ c2c4d log (2d3) . (6.143)

Similarily, if 1 + ε ≤ R(K) < 1 + 1/d then

ETNµ [R(Sρρρ(K))] ≤ 1 + ε (6.144)

whenever

N ≥log (d

ε)

log ( 4d4d−1

)≥

log ( dR(K)−1)

log ( 4d4d−1

)(6.145)

and

T ≥ c2c4d log (4d

ε) . (6.146)

It is clear that there exists a numerical contant c5 such that

log ( c2−1ε

)log ( 2d2

2d2−1)≤ c5d

2 log(1/ε) (6.147)

and

log (d/ε)log ( 4d

4d−1)≤ c5d log(1/ε). (6.148)

If we account for the first 3d Steiner symmetrizations needed to transform K into an

isomorphic ball see 6.133 then by using 6.141,6.144 as well as 6.147 and 6.148, we have

the following result:

Theorem 12. [9, Proposition 6.4]There exists a numerical constant c6 such that we need

at most⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

⌈c6d3 log(d) log(1/ε)⌉ if 1/d ≤ ε

⌈c6 (d3 log2(d) + d2 log2(1/ε))⌉ if ε < 1/d(6.149)

Steiner symmetrizations to transform any convex body K with volume κd into a body K ′

with the property R(K ′) ≤ 1 + ε.


6.4.2 Inradius

It is clear that relating Minkowski symmetrizations to Steiner symmetrizations will not

yield any information about the behaviour of the inner radius under consecutive Steiner

symmetrizations. We do, however, know that Steiner symmetrizations preserve volume

and we have a good handle on how many Steiner symmetrizations we need to reduce the

circumradius to an appropriate level. The fact that the convex body has volume equal to

that of the unit ball does not automatically yield any information about the inner radius

r(K) since we can have convex bodies that have very large widths in some directions

balanced with short widths in other directions. If we know beforehand that R(K) is

close to 1 then we cannot have very long widths and consequently we cannot have very

short widths either. This intuition is embodied in the following lemma.

Lemma 12. Let K be a convex body with volume κd. If R(K) ≤ 1+( ε24)d then r(K) ≥ 1−ε.

Proof. Temporarily let ε′ = ( ε24)d. The first step is to use Urysohn’s inequality (Proposi-

tion 24):

b(K)2

≥ 1. (6.150)

Now suppose there exists some u0 such that h(K,u0) < 1 − ε then since h(K,u) has

Lipschitz constant at most 1+ε′ we have h(K,w) < (1+ε′)(ε/4)+1−ε whenever ∣w−u0∣ ≤

ε/4. Let A = B(u0, ε/4) ∩ Sd−1 then

θ(A) = γ−1d ∫

proj(A)(1 − ∣x∣2)−1/2

dx > γ−1d Vd−1(proj(A)) (6.151)

with proj(A) the projection of A onto u⊥0 . We have

√∣x∣2 + (1 −

√1 − ∣x∣2)2 =

√2∣x∣ ≤ ε/4 (6.152)


whenever x ∈ Rd−1 and ∣x∣ ≤ ε4√

2. This implies

Vd−1(proj(A)) ≥ Vd−1 (ε

4√

2Bd−1) = ( ε

4√

2)d−1

κd−1. (6.153)

Combining 6.151 and 6.153 gives

θ(A) > κd−1

γd( ε

4√

2)d−1

= Γ(d/2 + 1)d√πΓ((d − 1)/2 + 1)

( ε

4√

2)d−1

> 1

d√π

( ε

4√

2)d−1

and thus

θ(A) > ( ε

(d√π)1/(d−1)4

√2)d−1

≥ ( ε

2√π4

√2)d−1

> ( ε

24)d−1

. (6.154)

We will now obtain a contradiction by showing that the mean width is smaller than 2.

We have

∫Sd−1

h(K,u)dθ(u) < θ(A)((1 + ε′)(ε/4) + 1 − ε) + (1 − θ(A))(1 + ε′)

< θ(A) (1 − ε2) + (1 − θ(A))(1 + ε′)

= 1 + ε′ − θ(A) ( ε2+ ε′)

< 1 + ε′ − θ(A) ε2

< 1.

6.4.3 Final Results

Proof of Theorem 6. If 0 < ε < 1 then ( ε24)d < 1/d. By Theorem 12, there exists a

numerical contant c6 such that we need at most

⌈c6 (d3 log2(d) + d4 log2(24/ε))⌉


Steiner symmetrizations to transform an arbitrary convex body K with volume κd into a

new convex body K ′ with the property R(K ′) ≤ 1+( ε24)d . By Lemma 12, r(K ′) ≥ 1−ε.

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convergence results for rearrangements: old and new. · pdf fileabstract convergence results...

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