A Courseon

Convex Geometry

Daniel Hug and Wolfgang WeilKarlsruhe Institute of Technology

revised version 2013

April 12, 2013

2

Preface

The following notes were written before and during the course onConvex Geometrywhich washeld at the University of Karlsruhe in the winter term 2002/2003. Although this was the firstcourse on this topic which was given in English, the materialpresented was based on previouscourses in German which have been given several times, mostly in summer terms. In comparisonwith these previous courses, the standard program was complemented by sections on surface areameasures and projection functions as well as by a short chapter on integral geometric formulas.The idea here was to lay the basis for later courses onStochastic Geometry, Integral Geometryetc., which usually follow in a subsequent term.

The exercises at the end of each section contain all the weekly problems which were handedout during the course and discussed in the weakly exercise session. Moreover, I have included afew additional exercises (some of which are more difficult) and even some hard or even unsolvedproblems. The list of exercises and problems is far from being complete, in fact the numberdecreases in the later sections due to the lack of time while preparing these notes.

I thank Matthias Heveling and Markus Kiderlen for reading the manuscript and giving hintsfor corrections and improvements.

Karlsruhe, February 2003 Wolfgang Weil

During repetitions of the course in 2003/2004 and 2005/2006a number of misprints and smallerrors have been detected. They are corrected in the currentversion. Also, additional materialand further exercises have been added.

Karlsruhe, October 2007 Wolfgang Weil

During the courses in 2008/2009 and 2010/2011 (by D. Hug) and2009/2010 (by W. Weil) theselecture notes have been revised and extended again. Also, some pictures have been included. Inthe winter term 2011/2012 a seminar on convex geometry (by B.Ebner and D. Hug) has beenbased on these lecture notes.

Karlsruhe, April 2013 Daniel Hug and Wolfgang Weil

3

4

Contents

Bibliography 7

Introduction 11

Preliminaries and notations 13

1 Convex sets 151.1 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 151.2 Combinatorial properties . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 221.3 Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 261.4 Support and separation theorems . . . . . . . . . . . . . . . . . . . .. . . . . . 301.5 Extremal representations . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 37

5

6 CONTENTS

Bibliography

[Al] A.D. Aleksandrov,Konvexe Polyeder. Akademie-Verlag, Berlin 1958.

[Ba] I.J. Bakelman,Convex Analysis and Nonlinear Geometric Elliptic Equations. SpringerBerlin et al. 1994.

[Bar] A. Barvinok,A Course in Convexity.AMS, Providence, RI 2002.

[Be] R.V. Benson,Euclidean Geometry and Convexity.McGraw-Hill, New York 1966.

[BKOS] M. de Berg, M. van Kreveld, M. Overmars, O. Schwarzkopf, Computational Geometry– Algorithms and Applications. Springer, Berlin, Second Revised Edition, 2000.

[Bl] W. Blaschke,Kreis und Kugel.2. Aufl., Walter der Gruyter, Berlin 1956.

[Boe] K. Boroczky Jr.,Finite Packing and Covering.Cambridge University Press, Cambridge2004.

[BY] J-D. Boissonnat, M. Yvinec,Algorithmic Geometry. Cambridge University Press 1998(English edition).

[BMS] V. Boltyanski, H. Martini, P.S. Soltan,Excursions into Combinatorial Geometry.Springer, Berlin et al. 1997.

[BF] T. Bonnesen, W. Fenchel,Theorie der konvexen Korper.Springer, Berlin 1934.

[BMP] P. Brass, W. Moser, J. Pach,Research Problems in Discrete Geometry.Springer, NewYork 2005.

[Bro] A. Brønsted,An Introduction to Convex Polytopes.Springer, Berlin et al. 1983.

[BZ] Y.D. Burago, V.A. Zalgaller,Geometric Inequalities.Springer, Berlin et al. 1988.

[Bu] H. Busemann,Convex Surfaces.Interscience Publ., New York 1958.

[Ed] H. Edelsbrunner,Algorithms in Combinatorial Geometry.Springer, Berlin 1987.

[Eg] H.G. Eggleston,Convexity.Cambridge Univ. Press, London et al. 1958.

7

8 BIBLIOGRAPHY

[Ew] G. Ewald,Combinatorial Convexity and Algebraic Geometry.Springer, New York et al.1996.

[FT] L. Fejes Toth,Lagerungen in der Ebene, auf der Kugel und im Raum.2. verb. u. erw.Aufl. Springer, Berlin 1972.

[Ga] R.J. Gardner,Geometric Tomography.Cambridge Univ. Press, Cambridge 1995. Revised2nd edition 2006.

[Go] J.E. Goodman, J. O’Rourke,Handbook of Discrete and Computational Geometry.CRCPress, Boca Raton, 1997.

[Gro] H. Groemer,Geometric Applications of Fourier Series and Spherical Harmonics.Cam-bridge Univ. Press 1996

[Gr] P.M. Gruber,Convex and Discrete Geometry.Grundlehren der mathematischen Wis-senschaften Bd. 336, Springer, Berlin 2007.

[Gru] B. Grunbaum,Convex Polytopes.Interscience Publ., London et al. 1967. 2. ed. (preparedby Volker Kaibel). Springer, New York, 2003.

[Gru2] B. Grnbaum, G.C. Shepard,Tilings and Patterns: an Introduction.Freeman, New York,1989.

[Ha1] H. Hadwiger,Altes und Neues uber konvexe Korper.Birkhauser, Basel et al. 1955.

[Ha2] H. Hadwiger,Vorlesungen uber Inhalt, Oberflache und Isoperimetrie.Springer, Berlin etal. 1957.

[HaDe] H. Hadwiger, H. Debrunner, V. Klee,Combinatorial Geometry in the Plane.Holt, Rine-hart and Winston, New York, 1964.

[Ho] L. Hormander,Notions of Convexity.Birkhauser, Basel et al. 1994.

[JoTh] M. Joswig, Th. Theobald,Algorithmische Geometrie: polyedrische und algebraischeMethoden.Vieweg, Wiesbaden, 2008.

[KW] L. Kelly, M.L. Weiss, Geometry and Convexity.Wiley/Interscience Publ., New York etal. 1979.

[Kl] R. Klein, Algorithmische Geometrie. Addison-Wesley-Longman, Bonn 1997.

[Ko1] A. Koldobsky,Fourier Analysis in Convex Geometry. Mathematical Surveys and Mono-graphs, American Mathematical Society, Providence RI 2005.

[KY] A. Koldobsky, V. Yaskin,The Interface between Convex Geometry and Harmonic Anal-ysis. CBMS Regional Conference Series in Mathematics, AmericanMathematical Soci-ety, Providence RI 2008.

BIBLIOGRAPHY 9

[Le1] K. Leichtweiß,Konvexe Mengen.Springer, Berlin et al. 1980.

[Le2] K. Leichtweiß,Affine Geometry of Convex Bodies.J.A. Barth, Heidelberg et al. 1998.

[Le] M. Leppmeier,Kugelpackungen von Kepler bis heute. Eine Einfuhrung furSchuler, Stu-denten und Lehrer.Vieweg, Braunschweig 1997.

[Ly] L.A. Lyusternik, Convex Figures and Polyhedra.Dover Publ., New York 1963.

[Ma] J.T. Marti,Konvexe Analysis.Birkhauser, Basel et al. 1977.

[Mat] J. Matousek,Lectures on Discrete Geometry. Graduate Texts in Mathematics, Vol. 212,Springer, New York, 2002.

[MS] P. McMullen, G.C. Shephard,Convex Polytopes and the Upper Bound Conjecture.Cam-bridge Univ. Press, Cambridge 1971.

[PA] J. Pach, P.K. Agarval,Combinatorial Geometry.Wiley-Interscience Series, Whiley, NewYork, 1995.

[Roc] R.T. Rockafellar,Convex Analysis.Princeton Univ. Press, Princeton 1970.

[Rog] C.A. Rogers,Packing and Covering.Cambridge University Press, Cambridge 1964.

[OR] J. O’Rourke,Computational Geometry in C. Cambridge University Press, Cambridge1994.

[S] R. Schneider,Convex Bodies: The Brunn-Minkowski Theory.Cambridge Univ. Press,Cambridge 1993.

[SW] R. Schneider, W. Weil,Integralgeometrie.Teubner, Stuttgart 1992.

[SW2] R. Schneider, W. Weil,Stochastische Geometrie.Teubner, Stuttgart 2000.

[SW3] R. Schneider, W. Weil,Stochastic and Integral Geometry.Springer, Berlin 2008.

[StW] J. Stoer, Ch. Witzgall,Convexity and Optimization in Finite Dimensions I.Springer,Berlin et al. 1970.

[Ths] R. R. Thomas,Lectures on Geometric Combinatorics.Lecture Notes, University ofWashington, Seattle 2004.

[Th] A.C. Thompson,Minkowski Geometry.Cambridge Univ. Press, Cambridge 1996.

[Va] F.A. Valentine,Convex Sets.McGraw-Hill, New York 1964. Deutsche Fassung:KonvexeMengen.BI, Mannheim 1968.

[We] R. Webster,Convexity.Oxford Univ. Press, New York 1964.

10 BIBLIOGRAPHY

[Zi] G.M. Ziegler, Lectures on Polytopes.Springer, Berlin et al. 1995. Revised 6th printing2006.

[Zo1] C. Zong,Strange Phenomena in Convex and Discrete Geometry.Springer, New York1996.

[Zo2] C. Zong,Sphere Packings.Springer, New York 1999.

[Zo3] C. Zong,The Cube: a Window to Convex and Discrete Geometry.Cambridge UniversityPress, Cambridg, 2006.

Introduction

Convexity is an elementary property of a set in a real (or complex) vector spaceV . A setA ⊂ Vis convex if it contains all the segments joining any two points ofA, i.e. if x, y ∈ A andα ∈ [0, 1]implies thatαx + (1 − α)y ∈ A. This simple algebraic property has surprisingly many andfar-reaching consequences of geometric nature, but it alsohas topological consequences (ifVcarries a compatible topology) as well as analytical ones (if the notion of convexity is extendedto real functions via their graphs). The interplay between convex sets and functions turns outto be particularly fruitful. Results on convex sets and functions play a central role in manymathematical fields, in particular in functional analysis,in optimization theory and in stochasticgeometry.

During this course, we shall concentrate on convex sets inRn as the prototype of a finite di-

mensional real vector space. In infinite dimensional spacesoften other methods have to be usedand different types of problems occur. Here, we concentrateon the classical part of convexity.Starting with convex sets and their basic properties (in Chapter 1), we briefly discuss convexfunctions (in Chapter 2), and then come (in Chapter 3) to the theory of convex bodies (com-pact convex sets). Our goal here is to present the essential parts of the Brunn-Minkowski theory(mixed volumes, quermassintegrals, Minkowski inequalities, in particular the isoperimetric in-equality) as well as some more special topics (surface area measures, projection functions). Inthe last chapter, we will shortly discuss selected basic formulas from integral geometry. If timepermits we will discuss symmetrization of convex sets and functions in an additional chapter.

The course starts rather elementary. Apart from a good knowledge of linear algebra (and, inChapter 2, analysis) no deeper knowledge of other fields is required. Later we will occasionallyuse results from functional analysis, in some parts, we require some familiarity with topologicalnotions and, more importantly, we use some concepts and results from measure theory.

11

12 INTRODUCTION

Preliminaries and notations

Throughout the course we work inn-dimensional Euclidean spaceRn. Elements ofRn aredenoted by lower case letters likex, y, . . . , a, b, . . . , scalars by greek lettersα, β, . . . and (real)functions byf, g, . . . We identify the vector space structure and the affine structure of Rn, i.e.we do not distinguish between vectors and points. The coordinates of a pointx ∈ R

n are usedonly occasionally, therefore we indicate them asx = (x(1), . . . , x(n)). We equipRn with its usualtopology generated by the standard scalar product

〈x, y〉 := x(1)y(1) + · · ·+ x(n)y(n), x, y ∈ Rn,

and the corresponding Euclidean norm

‖x‖ := ((x(1))2 + · · ·+ (x(n))2)1/2, x ∈ Rn.

By Bn we denote the unit ball,

Bn := {x ∈ Rn : ‖x‖ ≤ 1},

and bySn−1 := {x ∈ R

n : ‖x‖ = 1}

the unit sphere. Sometimes, we also make use of the Euclideanmetric d(x, y) := ‖x − y‖,x, y ∈ R

n. Sometimes it is convenient to writexα

instead of1αx, for x ∈ R

n andα ∈ R.Convex sets inR1 are not very exciting (they are open, closed or half-open, bounded or

unbounded intervalls), usually results on convex sets are only interesting forn ≥ 2. In somesituations, results only make sense, ifn ≥ 2, although we shall not emphasize this in all cases. Asa rule,A,B, . . . denote general (convex or nonconvex) sets,K,L, . . . will be used for compactconvex sets (convex bodies) andP,Q, . . . for (convex) polytopes.

A number of notations will be used frequently, without further explanations:

linA linear hull ofAaff A affine hull ofAdimA dimension ofA (= dimension ofaff A)intA interior ofArelintA relative interior ofA (interior w.r.t.aff A)clA closure ofAbdA boundary ofArelbdA relative boundary ofA

13

14 INTRODUCTION

If f is a function onRn with values inR or in the extended real line[−∞,∞] and if A isa subset of the latter, we frequently abbreviate the set{x ∈ R

n : f(x) ∈ A} by {f ∈ A}.HyperplanesE ⊂ R

n are therefore shortly written asE = {f = α}, wheref is a linear form,f 6= 0, andα ∈ R (note that this representation is not unique). The corresponding closed half-spaces generated byE are then{f ≥ α} and{f ≤ α}, and the open half-spaces are{f > α}and{f < α}.

The symbol⊂ always includes the case of equality. The abbreviation w.l.o.g. means ‘withoutloss of generality’ and is used sometimes to reduce the argument to a special case. The logicalsymbols∀ (for all) and∃ (exists) are occasionally used in formulas.� denotes the end of a proof.Finally, we write|A| for the cardinality of a setA.

Each section is complemented by a number of exercises. Some are very easy, but most requirea bit of work. Those which are more challenging than it appears from the first look are markedby ∗. Occasionally, problems have been included which are either very difficult to solve or evenunsolved up to now. They are indicated by P.

Chapter 1

Convex sets

1.1 Algebraic properties

In these lecture notes we usually identify an affine space with its associated vector space. How-ever, we shall deliberately use the words “vector” and “point”. The definition of a convex setrequires just the structure ofRn as an affine space. In particular, it should be compared with thenotions of a linear and an affine subspace.

Definition. A setA ⊂ Rn is convex, if αx+ (1− α)y ∈ A for all x, y ∈ A andα ∈ [0, 1].

Examples. (1)The simplest convex sets (apart from the points) are the segments. We denote by

[x, y] := {αx+ (1− α)y : α ∈ [0, 1]}

theclosed segmentbetweenx andy, x, y ∈ Rn. Similarly,

(x, y) := {αx+ (1− α)y : α ∈ (0, 1)}

is theopen segmentand we define half-open segments(x, y] and[x, y) in an analogous way.

(2) Other trivial examples are the affine flats inRn.

(3) If {f = α} (f 6= 0 a linear form,α ∈ R) is the representation of a hyperplane, theopenhalf-spaces{f < α}, {f > α} and theclosed half-spaces{f ≤ α}, {f ≥ α} are convex.

(4) Further convex sets are theballs

B(r) := {x ∈ Rn : ‖x‖ ≤ r}, r ≥ 0,

and their translates.

(5) Another convex set and a nonconvex set:

b

b

b

b

15

16 CHAPTER 1. CONVEX SETS

Let k ∈ N, let x1, . . . , xk ∈ Rn, and letα1, . . . , αk ∈ [0, 1] with α1 + . . . + αk = 1. Then

α1x1 + · · ·+ αkxk is called aconvex combinationof the pointsx1, . . . , xk.

Theorem 1.1.1.A setA ⊂ Rn is convex, if and only if all convex combinations of points inA lie

in A.

Proof. First, assume that all convex combinations of points inA lie in A. Takingk = 2 in thedefinition of a convex combination, we see that convexity ofA is implied.

For the other direction, assumeA is convex andk ∈ N. We use induction onk.Fork = 1, the assertion is trivially fulfilled.For the step fromk − 1 to k, k ≥ 2, assumex1, . . . , xk ∈ A andα1, . . . , αk ∈ [0, 1] with

α1 + . . .+ αk = 1. We may assume thatαk 6= 1. Then we define

βi :=αi

1− αk, i = 1, . . . , k − 1,

henceβi ∈ [0, 1] andβ1+. . .+βk−1 = 1. By the induction hypothesis,β1x1+. . .+βk−1xk−1 ∈ A.SinceA is convex, we conclude that

k∑

i=1

αixi = (1− αk)

(

k−1∑

i=1

βixi

)

+ αkxk ∈ A,

which completes the argument.

If {Ai : i ∈ I} is an arbitrary family of convex sets (inRn), then the intersection⋂

i∈I Ai isconvex. In particular, for a given setA ⊂ R

n, the intersection of all convex sets containingA isconvex, it is called theconvex hullconvA of A.

The following theorem shows thatconvA is the set of all convex combinations of points inA.

Theorem 1.1.2.For A ⊂ Rn,

convA =

{

k∑

i=1

αixi : k ∈ N, x1, . . . , xk ∈ A, α1, . . . , αk ∈ [0, 1],k∑

i=1

αi = 1

}

.

Proof. Let B denote the set on the right-hand side. IfC is a convex set containingA, Theorem1.1.1 impliesB ⊂ C. Hence, we getB ⊂ convA.

On the other hand, the setB is convex, since

β(α1x1 + · · ·+ αkxk) + (1− β)(γ1y1 + · · ·+ γmym)

= βα1x1 + · · ·+ βαkxk + (1− β)γ1y1 + · · ·+ (1− β)γmym,

for xi, yj ∈ A and coefficientsβ, αi, γj ∈ [0, 1] with α1 + . . .+ αk = 1 andγ1 + . . .+ γm = 1,and

βα1 + · · ·+ βαk + (1− β)γ1 + · · ·+ (1− β)γm = β + (1− β) = 1.

SinceB containsA, we getconvA ⊂ B.

1.1. ALGEBRAIC PROPERTIES 17

Remarks. (1)Trivially, A is convex, if and only ifA = convA.

(2) Later, in Section 1.2, we will give an improved version of Theorem 1.1.2 (CARATHEODORY’stheorem), where the numberk of points used in the representation ofconvA is bounded byn+1.

Definition. For setsA,B ⊂ Rn andα, β ∈ R, we put

αA+ βB := {αx+ βy : x ∈ A, y ∈ B}.

The setαA+βB is called alinear combination(or Minkowski combination) of the setsA,B, theoperation+ is calledvector addition(or Minkowski addition). Special cases get special names:

A+B thesum setA+ x (the caseB = {x}) a translateof AαA amultipleof AαA+ x (for α ≥ 0) ahomothetic imageof A−A := (−1)A thereflectionof A (in the origin)A− B := A+ (−B) thedifferenceof A andB

Remarks. (1)If A,B are convex andα, β ∈ R, thenαA+ βB is convex.

(2) In general, the relationsA + A = 2A andA− A = {0} arewrong. For a convex setA andα, β ≥ 0, we haveαA+ βA = (α+ β)A. The latter property characterizes convexity of a setA.

We next show that affine transformations preserve convexity.

Theorem 1.1.3.LetA ⊂ Rn, B ⊂ R

m be convex, and letf : Rn → Rm be an affine map. Then

f(A) := {f(x) : x ∈ A}

andf−1(B) := {x ∈ R

n : f(x) ∈ B}

are convex.

Proof. Both assertions follow from

αf(x) + (1− α)f(y) = f(αx+ (1− α)y).

Corollary 1.1.4. The projection of a convex set onto an affine subspace is convex.

The converse is obviously false, a shell bounded by two concentric balls is not convex but hasconvex projections.

Definition. (a) The intersection of finitely many closed half-spaces is called apolyhedral set.

(b) The convex hull of finitely many pointsx1, . . . , xk ∈ Rn is called a (convex)polytopeP .

18 CHAPTER 1. CONVEX SETS

(c) The convex hull of affinely independent points is called asimplex, anr-simplexis the convexhull of r + 1 affinely independent points.

Intuitively speaking, the vertices of a polytopeP form a minimal set of points fromP whichgenerate the polytope. A vertex of a polytopeP can also be characterized as a point ofP forwhichP \ {x} is still convex. We take the latter property as a definition ofa vertex.

Definition. A point x of a polytopeP is called a vertex ofP , if P \ {x} is convex. The set of allvertices ofP is denoted byvertP .

Theorem 1.1.5.LetP be a polytope inRn, and letx1, . . . , xk ∈ Rn be distinct points.

(a) If P = conv{x1, . . . , xk}, thenx1 is a vertex ofP , if and only ifx1 /∈ conv{x2, . . . , xk}.(b) P is the convex hull of its vertices.

Proof. (a) Let x1 be a vertex ofP . ThenP \ {x1} is convex andx1 /∈ P \ {x1}. Since{x2, . . . , xk} ⊂ P \{x1}, we getconv{x2, . . . , xk} ⊂ P \{x1}, and thusx1 /∈ conv{x2, . . . , xk}.

Conversely, assume thatx1 /∈ conv{x2, . . . , xk}. If x1 is not a vertex ofP , then there existdistinct pointsa, b ∈ P \ {x1} andλ ∈ (0, 1) such thatx1 = (1 − λ)a + λb. Hence there existk ∈ N, µ1, . . . , µk ∈ [0, 1] andτ1, . . . , τk ∈ [0, 1] with µ1 + . . . + µk = 1 andτ1 + . . .+ τk = 1such thatµ1, τ1 6= 1 and

a =

k∑

i=1

µixi, b =

k∑

i=1

τixi.

Thus we get

x1 =k∑

i=1

((1− λ)µi + λτi) xi,

from which it follows that

x1 =k∑

i=2

(1− λ)µi + λτi1− (1− λ)µ1 − λτ1

xi, (1.1)

where(1−λ)µ1+λτ1 6= 1 and the right-hand side of (1.1) is a convex combination ofx2, . . . , xk,a contradiction.

(b) Using (a), we can successively remove points from{x1, . . . , xk} which are not vertices with-out changing the convex hull. Moreover, ifx /∈ {x1, . . . , xk} and x is a vertex ofP , thenP = conv{x, x1, . . . , xk} implies thatx /∈ conv{x1, . . . , xk} = P , a contradiction.

Remarks. (1)A polyhedral set is closed and convex. Polytopes, as convex hulls of finite sets, areclosed and bounded, hence compact. We discuss these topological questions in more generalityin Section 1.3.

(2) For a polytopeP , Theorem 1.1.5 shows thatP = conv vertP . This is a special case ofM INKOWSKI ’s theorem, which is proved in Section 1.5.

(3) Polyhedral sets and polytopes are somehow dual notions. We shall see later in Section 1.4that the set of polytopes coincides with the set of bounded polyhedral sets.

1.1. ALGEBRAIC PROPERTIES 19

(4) The polytope property is preserved by the usual operations.In particular, ifP,Q are poly-topes, then the following sets are polytopes as well:

• conv(P ∪Q),

• P ∩Q,

• αP + βQ, for α, β ∈ R,

• f(P ), for an affine mapf : Rn → Rm.

Here, only the second assertion is not straight-forward. The proof thatP ∩ Q is a polytopewill follow later for instance from the mentioned connection between polytopes and boundedpolyhedral sets. The third assertion follows from Excercise 9.(5) If P is the convex hull of affinely independent pointsx0, . . . , xr, then eachxi is a vertex ofP , i.e.P is anr-simplex. Anr-simplexP has dimensiondimP = r.

Simplices are characterized by the property that their points are unique convex combinations ofthe vertices.

Theorem 1.1.6.A convex setA ⊂ Rn is a simplex, if and only if there existx0, . . . , xk ∈ A such

that eachx ∈ A has aunique representation as a convex combination ofx0, . . . , xk.

Proof. By definition, A is a simplex, ifA = conv{x0, . . . , xk} with affinely independentx0, . . . , xk ∈ R

n. The assertion therefore follows from Theorem 1.1.2 together with the unique-ness property of affine combinations (with respect to affinely independent points) and the well-known characterizations of affine independence (see also Exercise 11).

Exercises and problems

1. (a) Show thatA ⊂ Rn is convex, if and only ifαA+ βA = (α+ β)A holds, for allα, β ≥ 0.

(b) Which non-empty setsA ⊂ Rn are characterized byαA+βA = (α+β)A, for all α, β ∈ R?

2. LetA ⊂ Rn be closed. Show thatA is convex, if and only ifA+A = 2A holds.

3. A setR := {x+ αy : α ≥ 0}, x ∈ R

n, y ∈ Sn−1,

is called aray (starting inx with directiony).

LetA ⊂ Rn be convex and unbounded. Show thatA contains a ray.

Hint: Start with the case of a closed setA. For the general case, Theorem 1.3.2 is useful.

4. For a setA ⊂ Rn, thepolar A◦ is defined as

A◦ := {x ∈ Rn : 〈x, y〉 ≤ 1 ∀y ∈ A}.

Show that:

20 CHAPTER 1. CONVEX SETS

(a) A◦ is closed, convex and contains0.

(b) If A ⊂ B, thenA◦ ⊃ B◦.

(c) (A ∪B)◦ = A◦ ∩B◦.

(d) If P is a polytope,P ◦ is polyhedral.

5. (a) If ‖ · ‖′ : Rn → [0,∞) is a norm, show that the corresponding unit ballB′ := {x ∈ Rn :

‖x‖′ ≤ 1} is convex and symmetric (i.e.B′ = −B′).

(b) Show that

‖ · ‖1 : Rn → [0,∞), x = (x(1), . . . , x(n)) 7→

n∑

i=1

|x(i)|,

and‖ · ‖∞ : Rn → [0,∞), x = (x(1), . . . , x(n)) 7→ max

i=1,...,n|x(i)|,

are norms. Describe the corresponding unit ballsB1 andB∞.

(c) Show that for an arbitrary norm‖ · ‖′ : Rn → [0,∞) there are constantsα, β, γ > 0 such that

α‖ · ‖1 ≤ β‖ · ‖∞ ≤ ‖ · ‖′ ≤ γ‖ · ‖1.

Describe these inequalities in terms of the corresponding unit ballsB1, B∞, B′.

Hint: Show first the last inequality. Then prove that

inf{‖x‖∞ : x ∈ Rn, ‖x‖′ = 1} > 0,

and deduce the second inequality from that.

(d) Use (c) to show that all norms onRn are equivalent.

6. For a setA ⊂ Rn let

kerA := {x ∈ A : [x, y] ⊂ A for all y ∈ A}

be thekernelof A. Show thatkerA is convex. Show by an example thatA ⊂ B does not implykerA ⊂ kerB.

7. LetA ⊂ Rn be alocally finiteset (this means thatA ∩B(r) is a finite set, for allr ≥ 0). For each

x ∈ A, we define theVoronoi cell

C(x,A) := {z ∈ Rn : ‖z − x‖ ≤ ‖z − y‖ ∀y ∈ A},

consisting of all pointsz ∈ Rn which havex as their nearest point (or one of their nearest points)

in A.

(a) Show that the Voronoi cellsC(x,A), x ∈ A, are closed and convex.

(b) If convA = Rn, show that the Voronoi cellsC(x,A), x ∈ A, are bounded and polyhedral,

hence they are convex polytopes.

Hint: Use Exercise 3.

(c) Show by an example that the conditionconvA = Rn is not necessary for the boundedness of

the Voronoi cellsC(x,A), x ∈ A.

1.1. ALGEBRAIC PROPERTIES 21

8. Show that the setA of all convex subsets ofRn is a complete lattice with respect to the inclusionorder.

Hint: Define A ∧B := A ∩B,

A ∨B := conv(A ∪B),

infM :=⋂

A∈M

A, M ⊂ A,

supM := conv

(

⋃

A∈M

A

)

, M ⊂ A.

9. Show that, forA,B ⊂ Rn, we haveconv(A+B) = convA+ convB.

10. LetA,B ⊂ Rn be nonempty convex sets, and letx ∈ R

n. Show that

(a)conv({x} ∪A) = {λa+ (1− λ)x : λ ∈ [0, 1], a ∈ A}.

(b) If A ∩B = ∅, then

conv({x} ∪A) ∩B = ∅ or conv({x} ∪B) ∩A = ∅.

11. Assume thatx1, . . . , xk ∈ Rn are such that eachx ∈ conv{x1, . . . , xk} is a unique convex combi-

nation ofx1, . . . , xk. Show thatx1, . . . , xk are affinely independent.

12. LetP = conv{x0, . . . , xn} be ann-simplex inRn. Denote byEi the affine hull of{x0, . . . , xn} \{xi} and byHi the closed half-space bounded byEi and withxi ∈ Hi, i = 0, . . . , n.

(a) Show thatxi ∈ intHi, i = 0, . . . , n.

(b) Show thatP =

n⋂

i=0

Hi.

(c) Show thatP ∩ Ei is an(n− 1)-simplex.