convexity and asymptotic geometric...

What is the perimeter of a log–concave function?

Andrea Colesanti, in collaboration with Ilaria Fragala

Convexity and asymptotic geometric analysis

Centre de Recherches MathematiquesMontreal, 16–20 April 2012

Log–concave functions

We consider functions f : Rn → R+ of the form

f = e−u ,

whereu : Rn → (−∞,∞] is convex.

(with u 6≡ ∞). We will always assume

lim|x |→∞

f (x) = 0 ⇔ lim|x |→∞

u(x) =∞ .

Example.

f (x) = χK (x) =

{1 if x ∈ K ,0 if x 6∈ K ,

where K ⊂ Rn is a convex body. In this case

u(x) = IK (x) =

{0 if x ∈ K ,∞ if x 6∈ K ,

Log–concave functionsWe consider functions f : Rn → R+ of the form

f = e−u ,

(with u 6≡ ∞).

We will always assume

lim|x |→∞

f (x) = 0 ⇔ lim|x |→∞

u(x) =∞ .

Example.

f (x) = χK (x) =

{1 if x ∈ K ,0 if x 6∈ K ,

u(x) = IK (x) =

{0 if x ∈ K ,∞ if x 6∈ K ,

f = e−u ,

lim|x |→∞

f (x) = 0

⇔ lim|x |→∞

u(x) =∞ .

Example.

f (x) = χK (x) =

{1 if x ∈ K ,0 if x 6∈ K ,

u(x) = IK (x) =

{0 if x ∈ K ,∞ if x 6∈ K ,

f = e−u ,

lim|x |→∞

f (x) = 0 ⇔ lim|x |→∞

u(x) =∞ .

Example.

f (x) = χK (x) =

{1 if x ∈ K ,0 if x 6∈ K ,

u(x) = IK (x) =

{0 if x ∈ K ,∞ if x 6∈ K ,

f = e−u ,

lim|x |→∞

f (x) = 0 ⇔ lim|x |→∞

u(x) =∞ .

Example.

f (x) = χK (x) =

{1 if x ∈ K ,0 if x 6∈ K ,

u(x) = IK (x) =

{0 if x ∈ K ,∞ if x 6∈ K ,

f = e−u ,

lim|x |→∞

f (x) = 0 ⇔ lim|x |→∞

u(x) =∞ .

Example.

f (x) = χK (x) =

{1 if x ∈ K ,0 if x 6∈ K ,

where K ⊂ Rn is a convex body.

In this case

u(x) = IK (x) =

{0 if x ∈ K ,∞ if x 6∈ K ,

f = e−u ,

lim|x |→∞

f (x) = 0 ⇔ lim|x |→∞

u(x) =∞ .

Example.

f (x) = χK (x) =

{1 if x ∈ K ,0 if x 6∈ K ,

u(x) = IK (x) =

{0 if x ∈ K ,∞ if x 6∈ K ,

Operations on log–concave functions

Let f and g be log–concave and s > 0. We set

(f ⊕ g)(z) := supx+y=z

f (x)g(y) ,

(s · f )(x) := f s(x

I These operations preserve log–concavity: f ⊕ g and s · f arelog–concave.

I If K and L are convex bodies and t, s > 0, then

s · χK ⊕ t · χL = χsK+tL ,

wheresK + tL = {sx + ty : x ∈ K , y ∈ L} .

Let f and g be log–concave and s > 0.

We set

f (x)g(y) ,

(s · f )(x) := f s(x

f (x)g(y) ,

(s · f )(x) := f s(x

f (x)g(y) ,

(s · f )(x) := f s(x

f (x)g(y) ,

(s · f )(x) := f s(x

f (x)g(y) ,

(s · f )(x) := f s(x

The integral functional and its derivative

For f log–concave (verifying the decay condition at infinity), weconsider

I (f ) =

f (x) dx ∈ [0,∞) .

I If f = χK (K convex body), I (f ) = V (K ) (the volume of K ).

We are interested in the limit

δI (f , g) := limε→0+

I (f ⊕ ε · g)− I (f )

I If f = χK and g = χL (K and L convex bodies) then

δI (f , g) = limε→0+

V (K + εL)− V (K )

ε= n V (K , . . . ,K , L)

∫Sn−1

hL(ξ) dσK (ξ) =

∫∂K

hL(νK (x)) dHn−1(x) .

The integral functional and its derivativeFor f log–concave (verifying the decay condition at infinity), weconsider

I (f ) =

f (x) dx

∈ [0,∞) .

δI (f , g) := limε→0+

I (f ⊕ ε · g)− I (f )

V (K + εL)− V (K )

ε= n V (K , . . . ,K , L)

∫Sn−1

hL(ξ) dσK (ξ) =

∫∂K

I (f ) =

f (x) dx ∈ [0,∞) .

δI (f , g) := limε→0+

I (f ⊕ ε · g)− I (f )

V (K + εL)− V (K )

ε= n V (K , . . . ,K , L)

∫Sn−1

hL(ξ) dσK (ξ) =

∫∂K

I (f ) =

f (x) dx ∈ [0,∞) .

δI (f , g) := limε→0+

I (f ⊕ ε · g)− I (f )

V (K + εL)− V (K )

ε= n V (K , . . . ,K , L)

∫Sn−1

hL(ξ) dσK (ξ) =

∫∂K

I (f ) =

f (x) dx ∈ [0,∞) .

δI (f , g) := limε→0+

I (f ⊕ ε · g)− I (f )

V (K + εL)− V (K )

ε= n V (K , . . . ,K , L)

∫Sn−1

hL(ξ) dσK (ξ) =

∫∂K

I (f ) =

f (x) dx ∈ [0,∞) .

δI (f , g) := limε→0+

I (f ⊕ ε · g)− I (f )

I If f = χK and g = χL (K and L convex bodies)

V (K + εL)− V (K )

ε= n V (K , . . . ,K , L)

∫Sn−1

hL(ξ) dσK (ξ) =

∫∂K

I (f ) =

f (x) dx ∈ [0,∞) .

δI (f , g) := limε→0+

I (f ⊕ ε · g)− I (f )

V (K + εL)− V (K )

= n V (K , . . . ,K , L)

∫Sn−1

hL(ξ) dσK (ξ) =

∫∂K

I (f ) =

f (x) dx ∈ [0,∞) .

δI (f , g) := limε→0+

I (f ⊕ ε · g)− I (f )

V (K + εL)− V (K )

ε= n V (K , . . . ,K , L)

∫Sn−1

hL(ξ) dσK (ξ) =

∫∂K

I (f ) =

f (x) dx ∈ [0,∞) .

δI (f , g) := limε→0+

I (f ⊕ ε · g)− I (f )

V (K + εL)− V (K )

ε= n V (K , . . . ,K , L)

∫Sn−1

hL(ξ) dσK (ξ)

∫∂K

I (f ) =

f (x) dx ∈ [0,∞) .

δI (f , g) := limε→0+

I (f ⊕ ε · g)− I (f )

V (K + εL)− V (K )

ε= n V (K , . . . ,K , L)

∫Sn−1

hL(ξ) dσK (ξ) =

∫∂K

Representation formulas for δI (f , g) (1)

Let f = e−u, g = e−v . The equality

δI (f , g) =

v∗(∇u(x)) f (x) dx ,

where v∗ is the usual conjugate of v :

v∗(y) = supx∈Rn

x · y − v(x) ,

has been proved in some special cases.

I Klartag & Milman (2005), Rotem (2011): f = density of theGaussian measure.

I C. & Fragala (2011): general case, but with several additionalassumptions concerning smoothness, strict convexity, decay atinfinity of f and g .

I see also: Klartag (2007), who proved analogous results in thecontext of p–concave functions.

None of them includes the case: f = χK , g = χL, K and L convexbodies.

Representation formulas for δI (f , g) (1)Let f = e−u, g = e−v . The equality

δI (f , g) =

v∗(∇u(x)) f (x) dx ,

v∗(y) = supx∈Rn

x · y − v(x) ,

δI (f , g) =

v∗(∇u(x)) f (x) dx ,

v∗(y) = supx∈Rn

x · y − v(x) ,

δI (f , g) =

v∗(∇u(x)) f (x) dx ,

v∗(y) = supx∈Rn

x · y − v(x) ,

δI (f , g) =

v∗(∇u(x)) f (x) dx ,

v∗(y) = supx∈Rn

x · y − v(x) ,

δI (f , g) =

v∗(∇u(x)) f (x) dx ,

v∗(y) = supx∈Rn

x · y − v(x) ,

δI (f , g) =

v∗(∇u(x)) f (x) dx ,

v∗(y) = supx∈Rn

x · y − v(x) ,

δI (f , g) =

v∗(∇u(x)) f (x) dx ,

v∗(y) = supx∈Rn

x · y − v(x) ,

Let f = e−u and g = e−v and set

Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .

Sf and Sg are convex sets; let Hf and Hg be their supportfunctions. Then the equality

δI (f , g) =

v∗(∇u(x)) f (x) dx +

∫∂Sf

Hg (ν(x)) f (x) dHn−1(x) ,

where ν is the outer unit normal to Hf , has been proved underseveral additional assumptions on f , g , Sf and Sg .(C. & Fragala, 2011)

Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .

δI (f , g) =

v∗(∇u(x)) f (x) dx +

∫∂Sf

Hg (ν(x)) f (x) dHn−1(x) ,

Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .

Sf and Sg are convex sets; let Hf and Hg be their supportfunctions.

Then the equality

δI (f , g) =

v∗(∇u(x)) f (x) dx +

∫∂Sf

Hg (ν(x)) f (x) dHn−1(x) ,

Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .

δI (f , g) =

v∗(∇u(x)) f (x) dx +

∫∂Sf

Hg (ν(x)) f (x) dHn−1(x) ,

Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .

δI (f , g) =

v∗(∇u(x)) f (x) dx +

∫∂Sf

Hg (ν(x)) f (x) dHn−1(x) ,

where ν is the outer unit normal to Hf ,

has been proved underseveral additional assumptions on f , g , Sf and Sg .(C. & Fragala, 2011)

Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .

δI (f , g) =

v∗(∇u(x)) f (x) dx +

∫∂Sf

Hg (ν(x)) f (x) dHn−1(x) ,

where ν is the outer unit normal to Hf , has been proved underseveral additional assumptions on f , g , Sf and Sg .

(C. & Fragala, 2011)

Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .

δI (f , g) =

v∗(∇u(x)) f (x) dx +

∫∂Sf

Hg (ν(x)) f (x) dHn−1(x) ,

Perimeter

If K ⊂ Rn is a convex body, its perimeter, i.e. Hn−1(∂K ), can bedefined as

P(K ) = limε→0+

V (K + εB)− V (K )

where B is the unit ball of Rn.

I Define a corresponding notion of perimeter of a log-concavefunction f , as

P(f ) = limε→0+

I (f ⊕ ε · g)− I (f )

ε= δI (f , g) ,

with an appropriate choice of a unit ball g in the space oflog-concave functions.

Question: is there a natural candidate for g?

Perimeter

P(K ) = limε→0+

V (K + εB)− V (K )

P(f ) = limε→0+

I (f ⊕ ε · g)− I (f )

ε= δI (f , g) ,

Perimeter

P(K ) = limε→0+

V (K + εB)− V (K )

P(f ) = limε→0+

I (f ⊕ ε · g)− I (f )

ε= δI (f , g) ,

Perimeter

P(K ) = limε→0+

V (K + εB)− V (K )

P(f ) = limε→0+

I (f ⊕ ε · g)− I (f )

ε= δI (f , g) ,

Perimeter

P(K ) = limε→0+

V (K + εB)− V (K )

I Define a corresponding notion of perimeter of a log-concavefunction f ,

P(f ) = limε→0+

I (f ⊕ ε · g)− I (f )

ε= δI (f , g) ,

Perimeter

P(K ) = limε→0+

V (K + εB)− V (K )

P(f ) = limε→0+

I (f ⊕ ε · g)− I (f )

ε= δI (f , g) ,

Perimeter

P(K ) = limε→0+

V (K + εB)− V (K )

P(f ) = limε→0+

I (f ⊕ ε · g)− I (f )

ε= δI (f , g) ,

Perimeter

P(K ) = limε→0+

V (K + εB)− V (K )

P(f ) = limε→0+

I (f ⊕ ε · g)− I (f )

ε= δI (f , g) ,

The unit ball in the class of log–concave functions

I A natural possibility is to chose

g(x) = γn(x) = e−|x |2/2 ;

this choice was made also by Klartag & Milman and Rotem intheir definition of mean width of a log–concave function, andin [C. & Fragala, 2012].

I A drastically different choice is

g = χB = characteristic function of the unit ball of Rn,

which is what we consider here.

I There are clearly many other choices, (e.g.

g(x) = e−|x |p/p , p ≥ 1 .)

g(x) = γn(x) = e−|x |2/2 ;

g(x) = e−|x |p/p , p ≥ 1 .)

g(x) = γn(x) = e−|x |2/2 ;

g(x) = e−|x |p/p , p ≥ 1 .)

g(x) = γn(x) = e−|x |2/2 ;

g(x) = e−|x |p/p , p ≥ 1 .)

g(x) = γn(x) = e−|x |2/2 ;

g(x) = e−|x |p/p , p ≥ 1 .)

g(x) = γn(x) = e−|x |2/2 ;

I There are clearly many other choices,

g(x) = e−|x |p/p , p ≥ 1 .)

g(x) = γn(x) = e−|x |2/2 ;

g(x) = e−|x |p/p , p ≥ 1 .)

Definition of P(f ) and a useful formula

Let B be the unit ball of Rn. For a log–concave function f = e−u

define:

P(f ) := δI (f , χB) = limε→0+

I (f ⊕ ε · χB)− I (f )

Thm. (C. & Fragala, 2012) The function ε→ I (f ⊕ ε · χB) is apolynomial of degree n w.r.t. ε. Moreover, its (right) derivative atε = 0, i.e. P(f ), is

P(f ) =

∫ ∞0

[perimeter of the level set {f > t}] dt .

Note that {f > t} is convex for every t, by the log–concavity of f .

Let B be the unit ball of Rn.

For a log–concave function f = e−u

define:

P(f ) := δI (f , χB) = limε→0+

I (f ⊕ ε · χB)− I (f )

P(f ) =

∫ ∞0

define:

P(f ) := δI (f , χB) = limε→0+

I (f ⊕ ε · χB)− I (f )

P(f ) =

∫ ∞0

define:

P(f ) := δI (f , χB) = limε→0+

I (f ⊕ ε · χB)− I (f )

Thm. (C. & Fragala, 2012) The function ε→ I (f ⊕ ε · χB) is apolynomial of degree n w.r.t. ε.

Moreover, its (right) derivative atε = 0, i.e. P(f ), is

P(f ) =

∫ ∞0

define:

P(f ) := δI (f , χB) = limε→0+

I (f ⊕ ε · χB)− I (f )

P(f ) =

∫ ∞0

define:

P(f ) := δI (f , χB) = limε→0+

I (f ⊕ ε · χB)− I (f )

P(f ) =

∫ ∞0

define:

P(f ) := δI (f , χB) = limε→0+

I (f ⊕ ε · χB)− I (f )

P(f ) =

∫ ∞0

Perimeter and total variation

Let f = e−u : Rn → R be log–concave. By the previous result andcoarea formula, if moreover f ∈ C 1(Rn), we have

P(f ) =

∫ ∞0

[perimeter of the level set {f > t}] dt

|∇f | dx .

In the general case

P(f ) = |Df |(Rn) = total variation of f in Rn

|∇f | dx +

∫∂Sf

f dHn−1 .

These equalities confirm the validity of the representation formulasfor δI (f , g) seen before when f is arbitrary and g = χB .

Let f = e−u : Rn → R be log–concave.

By the previous result andcoarea formula, if moreover f ∈ C 1(Rn), we have

P(f ) =

∫ ∞0

|∇f | dx .

In the general case

|∇f | dx +

∫∂Sf

f dHn−1 .

Let f = e−u : Rn → R be log–concave. By the previous result andcoarea formula,

if moreover f ∈ C 1(Rn), we have

P(f ) =

∫ ∞0

|∇f | dx .

In the general case

|∇f | dx +

∫∂Sf

f dHn−1 .

P(f ) =

∫ ∞0

|∇f | dx .

In the general case

|∇f | dx +

∫∂Sf

f dHn−1 .

P(f ) =

∫ ∞0

|∇f | dx .

In the general case

|∇f | dx +

∫∂Sf

f dHn−1 .

P(f ) =

∫ ∞0

|∇f | dx .

In the general case

P(f ) = |Df |(Rn)

= total variation of f in Rn

|∇f | dx +

∫∂Sf

f dHn−1 .

P(f ) =

∫ ∞0

|∇f | dx .

In the general case

|∇f | dx +

∫∂Sf

f dHn−1 .

P(f ) =

∫ ∞0

|∇f | dx .

In the general case

|∇f | dx +

∫∂Sf

f dHn−1 .

P(f ) =

∫ ∞0

|∇f | dx .

In the general case

|∇f | dx +

∫∂Sf

f dHn−1 .

Properties of the perimeter

Recall:

P(f ) =

∫ ∞0

I Monotonicity:f ≤ g ⇒ P(f ) ≤ P(g) .

I Valuation property: for every f and g log–concave such thatf ∧ g is log–concave:

P(f ∧ g) + P(f ∨ g) = P(f ) + P(g) .

I Brunn-Minkowski inequality: for every f and g log–concave

P((1− t) · f ⊕ t · g) ≥ (P(f ))1−t (P(g))t .

This result was proved by Bobkov (2007).

Properties of the perimeterRecall:

P(f ) =

∫ ∞0

P(f ∧ g) + P(f ∨ g) = P(f ) + P(g) .

P((1− t) · f ⊕ t · g) ≥ (P(f ))1−t (P(g))t .

P(f ) =

∫ ∞0

P(f ∧ g) + P(f ∨ g) = P(f ) + P(g) .

P((1− t) · f ⊕ t · g) ≥ (P(f ))1−t (P(g))t .

P(f ) =

∫ ∞0

P(f ∧ g) + P(f ∨ g) = P(f ) + P(g) .

P((1− t) · f ⊕ t · g) ≥ (P(f ))1−t (P(g))t .

P(f ) =

∫ ∞0

P(f ∧ g) + P(f ∨ g) = P(f ) + P(g) .

P((1− t) · f ⊕ t · g) ≥ (P(f ))1−t (P(g))t .

P(f ) =

∫ ∞0

P(f ∧ g) + P(f ∨ g) = P(f ) + P(g) .

P((1− t) · f ⊕ t · g) ≥ (P(f ))1−t (P(g))t .

A Cauchy type formula

Let f be log–concave, ξ ∈ Sn−1 and Eξ = ξ⊥. We define theprojection of f onto Eξ as

fξ(x′) := sup

t∈Rf (x ′ + tξ) , x ′ ∈ Eξ .

Put simply, the graph of fξ is the projection of the graph of f ontoEξ × R. Let

In−1(fξ) :=

∫Eξ

fξ(x′) dx ′ .

P(f ) = c(n)

∫Sn−1

In−1(fξ) dHn−1(ξ) .

Let f be log–concave, ξ ∈ Sn−1 and Eξ = ξ⊥.

We define theprojection of f onto Eξ as

fξ(x′) := sup

t∈Rf (x ′ + tξ) , x ′ ∈ Eξ .

In−1(fξ) :=

∫Eξ

fξ(x′) dx ′ .

P(f ) = c(n)

∫Sn−1

fξ(x′) := sup

t∈Rf (x ′ + tξ) , x ′ ∈ Eξ .

In−1(fξ) :=

∫Eξ

fξ(x′) dx ′ .

P(f ) = c(n)

∫Sn−1

fξ(x′) := sup

t∈Rf (x ′ + tξ) , x ′ ∈ Eξ .

Put simply, the graph of fξ is the projection of the graph of f ontoEξ × R.

In−1(fξ) :=

∫Eξ

fξ(x′) dx ′ .

P(f ) = c(n)

∫Sn−1

fξ(x′) := sup

t∈Rf (x ′ + tξ) , x ′ ∈ Eξ .

In−1(fξ) :=

∫Eξ

fξ(x′) dx ′ .

P(f ) = c(n)

∫Sn−1

fξ(x′) := sup

t∈Rf (x ′ + tξ) , x ′ ∈ Eξ .

In−1(fξ) :=

∫Eξ

fξ(x′) dx ′ .

P(f ) = c(n)

∫Sn−1

Isoperimetric inequalities

I Using the definition of P(f ) and Prekopa–Leindler inequalitywe find:

P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)

where: Ent(f ) =

f log(f ) dx − I (f ) log(I (f )) ,

and equality holds iff f = χB (up to a translation).

I The Sobolev inequality in Rn for functions of boundedvariation yelds

P(f ) ≥ nω1/nn ‖f ‖ n

n−1, (2)

where ωn is the volume of the unit ball. Equality holds ifff = χB (up to translation).

P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)

where: Ent(f ) =

P(f ) ≥ nω1/nn ‖f ‖ n

n−1, (2)

P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)

where: Ent(f ) =

P(f ) ≥ nω1/nn ‖f ‖ n

n−1, (2)

P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)

where: Ent(f ) =

P(f ) ≥ nω1/nn ‖f ‖ n

n−1, (2)

P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)

where: Ent(f ) =

P(f ) ≥ nω1/nn ‖f ‖ n

n−1, (2)

P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)

where: Ent(f ) =

P(f ) ≥ nω1/nn ‖f ‖ n

n−1, (2)

P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)

where: Ent(f ) =

P(f ) ≥ nω1/nn ‖f ‖ n

n−1, (2)

where ωn is the volume of the unit ball.

Equality holds ifff = χB (up to translation).

P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)

where: Ent(f ) =

P(f ) ≥ nω1/nn ‖f ‖ n

n−1, (2)

convexity and asymptotic geometric...

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