Krein Milman theorem and its applications

Krein-Milman Theorem

and its applications

Bachelor’s degree final project

Author:

Antonio Zarauz Moreno

Advisor:

Dr. Juan Carlos Navarro Pascual

Degree in Mathematics

CIENCIAS EXPERIMENTALES

LOGOTIPO EN COLOR

UNIVERSIDAD DE ALMERÍA

July, 2016

Contents

1 Basic concepts and finite-dimensional theory 1

1.1. Convex sets and maps 1

1.2. Carathéodory’s theorem 6

1.3. Dimension of a convex set 10

1.4. Extreme points 12

1.5. Carathéodory-Minkowski theorem 15

2 Krein-Milman theorem 19

2.1. Examples of extreme points 19

L_p([0,1]) spaces, 20.— c0 space, 21.—`1 space, 21.—`∞ space, 22.—C(X) space, 23.— Probability measures on a compact Hausdorffspace,23.— Characterisation of strictly con-vex spaces by extreme points,26.

2.2. Characterisation of unit balls in normed spaces 27

2.3. Krein-Milman theorem 29

2.4. Applications of Krein-Milman theorem 33

Necessary condition for duality,33.— Representation theorem,34.— The Stone-Cech compactification,

35.— Banach-Stone theorem,37.— Stone-Weierstrass theorem,39.

Bibliography 41

Abstract in English

The present degree’s project intends to be a brief guide for those who want to be-gin a path in convex set theory and functional analysis. This ambitious project has also been motivated by the achievement of Collaboration’s scholarship for the student and it is a first approach to the contents that have been developed during the whole academic course.

The title of this work, Krein-Milman theorem and its applications, contains the es-sence of our purpose, which is delving into the previous theorem and its requirements: topological vector spaces, convexity, measure theory, and so on and so forth. The ref-erence of the original paper can be consulted in [1].

The first chapter introduces clearly the essential concepts of convex analysis, and also the main ideas of this topic in finite-dimensional spaces, due to Carathéodory and Minkowski. In that environment, the writer has included some graphic examples (and counterexamples) which are supposed to be profitable for the reader. The structure of the chapter and the obtaining of the main results has been elaborated via [2], [3] and [4]. Specifically:

In the first section we have used [2] to introduce the definitions, whereas the rest of result has been developed by the student.

In the second section lies one of the most important theorems of the chapter (Carathéodory’s theorem), and it also incorporates a long list of remarks to show the importance and the improvable facts of the theorem. In the following section, we discuss the notion of dimension of a convex set and its relative interior. To write these sections, the writer has selected [4].

To conclude the chapter, sections four and five comprise concepts of paramount importance for the next chapter, such as extreme point, face or exposed point. One can highlight the canonical way to build faces via continuous linear func-tionals, the existence of extreme points for compact convex sets, the Carathéodory-Minkowski’s theorem and one of the most beautiful applications in finite-dimensional theory, which is the existence of extreme values of linear functionals over com-pact convex sets in extreme points of the domain. To elaborate this sections we have consulted [3].

The second chapter is devoted to the exposition of Krein-Milman theorem, giving a wide introduction to the infinite-dimensional spaces via several examples of canonical spaces. Specifically,

The enriching list of examples of extreme points has been developed in order to get the reader used to the main strategies which lies into this theory; the sixth example is inspired by [9] and [13].

Before getting into Krein-Milman theorem, there are some considerations about the origin of the main algebraic concepts involved in this theory, which are bal-ancedness, absorbency and the own definition of convexity, through the algebraic

characterisation of unit balls in seminormed spaces (it has been required [14]). Furthermore, since compactness is one of the most cultivated concepts in Gen-eral Topology, it is reasonable to study the convex hull of a compact set, giving a solution to that problem in an infinite-dimensional context.

The proof of the mentioned theorem is based on [6]. Some interesting remarks are made after the theorem, which delve into the conditions applied on the the-orem and the notation of some other authors. Finally, it will be discussed one more detail about compactness, which reflects the fact that those sets contain every extreme point of the closure of their convex hull, i.e., Milman theorem (also based on [6]).

Resumen en español

El presente trabajo de fin de grado pretende ser una breve guía para aquellos que quieren dar unos primeros pasos en la teoría de conjuntos convexos y análisis fun-cional. Este ambicioso proyecto ha sido motivado por la obtención de la beca de Col-aboración por parte del alumno, y es una primera aproximación a los contenidos que han sido desarrollados durante el curso académico.

El título de este trabajo, El teorema de Krein-Milman y sus aplicaciones, contiene la esencia de nuestro objetivo, que consiste en analizar en profundidad el anterior teorema y todos sus prerrequisitos: espacios vectoriales topológicos, convexidad, teoría de la medida, etcétera. La referencia original puede ser consultada en [1].

El primer capítulo introduce de forma clara los conceptos esenciales del análisis convexo, y también las principales ideas de esta materia en espacios finito-dimensionales, debidos a Carathéodory y Minkowski. En este contexto, el se han incluido numerosos ejemplos gráficos (y contraejemplos) que serán productivos para el lector. La estruc-tura del capítulo y la obtención de los principales resultados ha sido elaborada a partir de [2], [3] and [4]. Concretamente:

En la primera sección se ha usado [2] para introducir definiciones, mientras que el resto de resultados han sido desarrollados por el estudiante.

La segunda sección alberga uno de los teoremas más importantes del capítulo (teorema de Carathéodory) y también incorpora una larga lista de observaciones para ensalzar su importancia y los detalles mejorables del mismo. En la siguiente sección, se discute la noción de dimensión de un conjunto convexo y su interior relativo. Para escribir estas secciones, se ha escogido [4].

Para concluir el capítulo, las secciones cuatro y cinco comprenden conceptos de primordial importancia para el siguiente capítulo, tales como punto extremo, cara de un conjunto convexo o punto expuesto. Resaltamos además la forma canónica de construir caras mediante funcionales lineales y continuos, la exist-encia de puntos extremos para conjuntos compactos y convexos, el teorema de Carathéodory-Minkowski y una de las más relucientes aplicaciones en un con-texto finito-dimensional, que es la existencia de valores extremos en funcionales lineales y continuos sobre compactos convexos. Para elaborar estas secciones, hemos consultado [3].

El segundo capítulo está dedicado a la exposición del teorema de Krein-Milman, dando una amplia introducción a los espacios infinito-dimensionales mediante ejemplos sobre espacios canónicos. Concretamente,

La enriquecedora lista de ejemplos sobre puntos extremos ha sido desarrollada para introducir al lector en las principales estrategias que subyacen en esta teoría; el sexto ejemplo está inspirado en [9] y [13].

Antes de adentrarnos en el teorema de Krein-Milman, hacemos algunas consid-eraciones sobre el origen de las principales definiciones algebraicas tratadas en

el tema, tales como equilibrio, absorbencia y el propio concepto de convexidad, a través de la caracterización algebraica de bolas unidad en espacios seminor-mados (ha sido requerido [14]). Además, dado que la compacidad es una de las más cultivadas de la Topología General, es razonable estudiar la envolvente con-vexa de conjuntos compactos, dando una solución contundente a tal problema en espacios de dimensión infinita.

La demostración del mencionado teorema está basada en [6]. Algunas observa-ciones oportunas son propuestas tras el teorema, que profundizan en las con-diciones aplicadas en la hipótesis del teorema y la notación de algunos autores. Finalmente, será considerado un detalle adicional sobre compacidad que refleja el hecho de que tales conjuntos contienen cada puntos extremo del cierre de su envolvente convexa, i.e., el teorema de Milman (también basado en [6]).

1

Basic concepts and finite-dimensional

theory

The first chapter will be devoted to the exposition of several elementary notions related to convex and functional analysis. We start our path to Krein-Milman the-orem proving its famous finite-dimensional preceding; i.e., Carathéodory-Minkowski theorem.

Recall that a topological vector space is a pair (X, τ) whereX is a vector space over the fieldK=R∨C, andτis a compatible topology with the vector structure inX; that

is, the maps (x, y)7→_{x+yand (α, x)}7→_{αxare continuous fromX}×_{XontoXand from}

K×X ontoXrespectively, considering the product topology in each space.

Secondly, a normed space is a pair (X,k·k_{) whereX is a vector space and}k·k_{a norm} inX. Since the topology induced by the norm is compatible with the vector structure, normed space form a strongly relevant example of topological vector spaces. There also are other structures which are compatible with the norm, such as the weak topo-logy of a normed space X, denoted by ω, and the weak-star topology, written asω∗. As usual, we write X instead of (X, τ) or (X,k·k_{) when we are making reference to a} topological vector space or a normed space, respectively.

Letnbe a natural number andXa Hausdorfftopological vector space with dim(X) =

n. Then, every linear bijection fromKnontoXis bicontinuous, henceX is isomorphic

as a vector space to Kn and homeomorphic as a topological vector space to the

Eu-clidean space. However, the notation X for finite-dimensional spaces will be used during the whole chapter, since it will make easier the step of abstraction given in the following chapters.

1.1

Convex sets and maps

Definition 1.1. In a vector spaceX overK, a subsetA⊂Xis convex if, given x, y∈Aand

t∈_[0,1],

{_{tx+ (1}−_t)y:t∈_]0,1[} ⊂_{A. (1.1)}

Definition 1.2. IfA⊂_{X is a convex set, a function f :A}→_{Ris said to be convex (resp.}

concave) if the following inequality holds for eachx, y∈_Aandt∈_[0,1]:

f(tx+ (1−_t)y)≤_{tf(x) + (1}−_{t)f(y) (resp.f(tx+ (1}−_t)y)≥_{tf(x) + (1}−_{t)f(y)). (1.2)}

If the inequalities 1.2 are strict,f is strictly convex (resp. strictly concave), and if it is an

equality in both cases,f is affine.

The following examples can be easily checked by the reader.

Examples 1.1.

Any segment (either open or closed) is a convex set:

Sx,y ={tx+ (1−t)y: t∈[0,1]},

◦

Sx,y ={tx+ (1−t)y: t∈]0,1[}

wherex, yare arbitrary points of a vector space.

Any hyperplane is a convex set

H={_x∈_{X: f(x) =λ}}_,

where f :X →_{Ris a linear functional on a vector space and λ}∈ _{R, and every}

half-space

H1={x∈X:f(x)≤λ}, H2 ={x∈X:f(x)≥λ} is also a convex set.

Any ball (either open or closed) in a normed spaceXis a convex set

B(x, r) ={_y∈_X: k_y−_xk_{< r}}_{, B(x, r) =}{_y∈_X:k_y−_xk_{< r}}_,

Actually, every set with the formC =F∪_{B(x, r), F}⊂_{∂B(x, r) :=S(x, r)is convex.}

LetXbe a vector space and considerA⊂_{X. The convex hull ofAis the intersection of}

all the convex subsets ofXcontainingA. It is clear thatA⊂_{co(A)and co(A)is convex}

(see proposition1.1); in addition, we state that co(A)admits the expression

co(A) =       

n

X

i=1

λixi: n∈N, xi∈A, λi∈R+0,∀i ={1, . . . , n},

n

X

i=1

λi = 1

      

, (1.3)

In fact, letV be the set in the right side of the equation (1.3).

To verify that co(A)⊂_{V it is enough to show thatV is convex (note thatA}⊂_{V). Given}

a=Pn

i=1αiyi andb=Pmi=1βizi elements inV (suppose without loss of generality that

n≤_{m), for eacht}∈_{[0,1]we make the change of variables}

λi =

(

(1−_{t)αi if i = 1, . . . , n}

tβi if i =n+ 1, . . . , n+m , xi =

(

(1−_{t)yi if i= 1, . . . , n}

so we obtain (1−_{t)a+tb =}Pn+m

i=1 λixi where Pin=1+mλi = 1 and xi ∈ A for every i =

1, . . . , n+m.

The other inclusion will be proved by induction. For n = 1 it is clear, so long as

A⊂_{co(A)and co(A)is a convex set. Suppose that the statement holds forn}∈_Nand

letx=Pn+1

i=1λixi. Ifλn+1= 0∨λn+1= 1is straightforward. Otherwise,

x= (1−_λn+1)

n

X

i=1

λi

1−_λn+1xi+λn+1xn+1,

which is a convex combination of Pn

i=1

λi

1−_λn+1xi and xn+1, both elements of Aby

in-duction hypothesis.

Figure 1.1: Convex hull of a galleon.

In particular, if A is a finite union of convex sets; i.e., A =∪n

i=1Ai with Ai convex

for every i = 1, . . . , n, then one can choose every point of the previous convex linear

combination in eachAi:

co(A) =        n X i=1

λixi, xi∈Ai, λi ∈R+0,∀i∈ {1, . . . , n},

n

X

i=1

λi= 1

       . (1.4)

First of all, the reader should appreciate that the numbernis fixed under these

circum-stances. It is easy to check that the set

E=        n X i=1

λixi, xi∈Ai, λi ∈R+0,∀i∈ {1, . . . , n},

n

X

i=1

λi = 1

      

is convex. On the one hand, the inclusion A⊂_{E and the convexity of E implies that}

co(A)⊂_{E. On the other hand, the previous example shows thatE}⊂_co(A).

In a similar way, we define the real affine hull ofA⊂_{X as}

aff(A) =        n X i=1

λixi: n∈N, xi∈A, λi∈R,∀i∈ {1, . . . , n},

n

X

i=1

λi= 1

and it also verifies that it is the least affine space which contains A, and aff(A) =

aff(co(A)).

Every affine map is linear-convex by definition. Reciprocally, we can suppose thatt <0

without losing generality (ift >1we can interchange the role ofxandy). Then,

f(y) =f

                1 1−_t |{z}

∈[0,1]

(tx+ (1−_{t)y) +}

1− 1 1−_t

x                 .

Using the affinity off, this reduces to

f(y) = 1

1−_tf(tx+ (1−t)y) +

1− 1 1−_t

f(x)⇒_{f(tx+ (1}−_{t)y) =tf(x) + (1}−_t)f(y).

Proposition 1.1. LetC_{the family of all the convex sets of a vector spaceX. Then,}

1. Whatever{_Ci}_i∈I ⊂ C_is∩_i∈ICi∈ C_.

2. C _{satisfies thatA+B}∈ C_,λA∈ C _{for allA, B}∈ C _andλ∈_{R. In addition,(λ+µ)A=}

λA+µAfor everyλ, µ∈_{Rsuch thatλµ}≥_0.1

3. co(·_{) :X}→ C_{is a monotone and additive operator.}

4. Ais convex iffA=co(A).

Proof. 1. Givenx, y∈ ∩_{i∈ICi, sincex, y}∈_{Ci for alli} ∈_{I,tx+ (1}−_t)y∈_{Ci for alli} ∈_{I and}

t∈_{[0,1]. Hencetx+ (1}−_t)y∈ ∩_{i∈ICi for everyt}∈_[0,1].

2. Let x, y∈_{A+Band t}∈_{[0,1]. We can expressx=ax+bx, y =ay+by withax, ay} ∈

A, bx, by ∈B. Then,

tx+(1−_{t)y=t(ax+bx)+(1}−_{t)(ay+by) = [tax+(1}−_{t)ay]+[tbx+(1}−_t)by]∈_A+B,∀_t∈_[0,1]. Furthermore, for everyλ∈_{Randx, y}∈_A,

t(λx) + (1−_{t)(λy) =λ[tx+ (1}−_t)y]∈_λA,∀_t∈_[0,1].

The last statement is checked as follows: the implication (λ+µ)A⊂_{λA+µAis clear} thanks to the distributive law inK; reciprocally, it is straightforward whenλ= 0∨µ=

0. Otherwise, letλa∈_{λA, µa}0∈_{µA, then}

λa+µa0= λ

λ+µ(λ+µ)a+ µ

λ+µ(λ+µ)a 0

.

Since the previous equality is a convex combination of elements in (λ+µ)A, the result lies in that set, henceλA+µA⊂_(λ+µ)A.

1_{Indeed, one can characteriseC through this property, in an even easier way than the implication}

we have already proved. In fact, ifAis not convex, we can findx∈_X,λ=µ=1

2 satisfying thatx∈Abut

3. The monotony derives from the definition. To see that co(A+B)⊂_{co(A) + co(B),} given x ∈ _{co(A+B), there exist} {_λi}k

i=1 ⊂ R +

0, {ai}k_i=1 ⊂ A and {bi}k_i=1 ⊂ B such that

Pk

i=1λi = 1 and

x=

k

X

i=1

λi(ai+bi) = k

X

i=1

λiai+ k

X

i=1

λibi ∈co(A) + co(B).

On the other hand, given a ∈ _{co(A) and b} ∈ _{co(B), there exists} {_λi}k

i=1,{µj}mj=1 ⊂ R+0,

{_ai}k

i=1⊂Aand{bj}mj=1⊂Bsuch that

Pk

i=1λi =Pmj=1µj= 1 and

a+b=

m X j=1 µj |{z} 1        k X i=1

λiai

       + k X i=1 λi |{z} 1         m X j=1

µjbj

        =X i,j

λiµj(ai+bj)∈co(A+B)

sinceP

i,jλiµj =

_Pk

i=1λi

Pm

j=1µj

= 1 andλiµj≥0 for eachi, j.

4. ⇒_{) SinceAis convex andA}⊂_{A, we have that co(A)}⊂_A. ⇐_{) It is clear that co(A) is convex, henceAis convex too.}

Remarks 1.1.

The union of convex sets is not necessarily a convex set, as the next figure based on

examples1.1shows:

The intersection of hyperplanes gives us the solution of a linear system of equations, and the intersection of half-spaces gives us a polyhedron; a bounded polyhedron is

called a polytope. In particular, the k-simplex (k ≤ _{N + 1) determined by a set of}

affine-independent points{_xi}k

i=1⊂Xis∆k

{_xi}k

i=1

=co{_xi}k

i=1

(figure1.3).

Figure 1.3: Example of 4-simplex inX=R3(tetrahedron).

1.2

Carathéodory’s theorem

LetXbe a vector space. The definition of convex hull of a subsetA⊂_{Xbrings us a} characterisation of convex sets through proposition1.1. However, there is no limit in the number of elements involved in the representation of eachx∈_{co(A). In this sense,} Carathéodory’s theorem states that every point x∈ _{co(A) can be expressed withn+ 1} points ofAas much.

The following lemma shows, in particular, the highest number of linear-independent elements.

Lemma 1.1. A set of points{_xi}k

i=1⊂X is affine-dependent iffthere exists{λi}ki=1∈Rsuch

thatPk

i=1λi= 0<Pki=1|λi|andPki=1λixi = 0.

Proof. Since{_xi}k

i=1⊂Xis affine-dependent, we have that{xj−x1}kj=2is linear-dependent

as a set of vectors, so there exists{_αj}k

j=2with

Pk

j=2|αj|>0 and

0 =

k

X

j=2

αj(xj−x1) =

        −

k

X

j=2

αj

       

x1+

k

X

j=2

αjxj

(

λ1 = −Pkj=2αj

λj = αj, j= 2, . . . , k

! =

k

X

i=1

λixi.

The collection of numbers{_λi}k

Theorem 1.1 (Carathéodory). If dim(X) = n, A ⊂ _{X and x} ∈ _{co(A), then x is a convex}

combination of affine-independent points fromA(in particular,n+ 1as much).

Proof. Letx∈_{co(A) such that}

x=

k

X

i=1

λixi

with{_xi}k

i=1⊂A,{λi}ki=1⊂R+0 and

Pk

i=1λi = 1 to be the shortest expression ofxin terms

of elements of A. Byreductio ad absurdum, suppose that{_xi}k

i=1 are affine-dependent.

The previous lemma1.1shows that there exists{_αi}k

i=1⊂Rsatisfying

k

X

i=1

αi = 0< k

X

i=1

|_αi|_,

k

X

i=1

αixi = 0.

It can also be considered, without losing generality, that

λk

αk

= min

i=1,...,k

(

λi

αi

:αi>0

)

,

the objective now is looking for a linear combination of x in terms of{_xi}k−1

i=1 to find a

contradiction:

x=

k

X

i=1

λixi = k−₁

X

k=1

λi−

λk

αk

αi

!

xi+ k−₁

X

i=1

λk

αk

αixi+λkxk

| {z }

0

=

k−₁

X

k=1

λi−

λk

αk

αi

!

xi.

Calling ξi = λi − αλkkαi for each i = 1, . . . , k

−_{1, it is clear that ξi} ≥ _{0 because of the} assumption over λk

αk. Finally, k−₁

X

i=1

ξi =

      

λk+ k−₁

X i=1 λi        −       

λk+ k−₁

X i=1 λk αk αi       

= 1−λk

αk k X i=1 αi |{z} 0

= 1.

In spite of its usefulness, this result does not give any information about the points we select to express somex∈_{co(A). In fact, as we have already seen, this result is valid} for every vector space (with no topological structure). However, it would be desirable to obtain a more powerful result with the aid of a suitable structure.

The next step in the process will be the choice of areducedgroup of points P of a setAsatisfying co(P) =A. That is a first approach to what Krein-Milman theorem will state in next chapter for locally convex topological vector spaces:

co(P) =A.

Convexity’s hypothesis is clear, since our purpose is thereconstructionof the setA

through its convex hull (or its closed convex hull if necessary) of a distinguished subsetP ofA.

Compactness is also required, since one can find examples of closed and bounded convex sets which has no extreme points (see subsection2.1). As well as in many other branches of Mathematics, this is a convenient hypothesis to ensure the ex-istence of the previous setP ⊂_{Asatisfying the desired condition.}

In the rest of the section, we will assume thatXis a finite-dimensional topological (only required for proposition1.2) vector space; this is enough to prove Carathéodory-Minkowski theorem, even if the previous results can be discussed in more general structures.

The next results show the topological properties of co(·_{) as an operator over}C_.

Lemma 1.2. GivenA⊂_Xconvex,x∈_int(A)andy∈_{A, is}

◦

Sx,y⊂int(A).

Proof. Lett∈_{]0,1[ be fixed; we have to show thattx+ (1}−_t)y∈_{int(A). By translation if}

necessary we can assume thattx+ (1−_{t)y= 0, in particulary=αxwhereα <0. Since} the mapping ω 7→_{αω is a homeomorphism of X and x} ∈ _{int(A), y} ∈ _{A, there exists}

z∈_{int(A) such thatαz}∈_A. Letµ= _αα−₁; thenµ∈]0,1[ and

µz+ (1−_{µ)αz= 0.} Then, the set

U ={_{µω+ (1}−_µ)αz:ω∈_int(A)}

is a 0-neighbourhood, as long asω7→_µω+(1−_{µ)αzis a homeomorphism ofXmapping}

z∈_{int(A) onto 0. But ω}∈_{int(A) andαz}∈_{Aimply that U} ⊂_{Afor beingAconvex, and} 0∈_int(A).

Proposition 1.2. Given a setA⊂_{X, we have:}

1. int(A)andAare convex sets ifAis convex.

2. co(·_{)maps open sets into open sets.}

3. co(·_{)maps bounded sets into bounded sets.}

4. co(·_{)maps compact sets into compact sets.}2

5. co(·_{)maps precompact sets into precompact sets.}

6. IfAis convex and int(A),∅, then int(A) =int(A)and int(A) =A.

Proof. 1. For anyx, y∈_{int(A), we have that}

◦

Sx,y ⊂int(A) by lemma1.2, soSx,y ⊂int(A)

and int(A) is convex. Furthermore, givenx, y ∈_{A andt} ∈_{[0,1], there exist sequences}

{_xn}_n∈

N,{yn}n∈N in A which converge to x and y respectively. By convexity of A, the

family of sequences

zn,t=txn+ (1−t)yn, n∈N,

belong to A and converge to tx+ (1−_{t)y for each t} ∈ _{[0,1]. HenceSx,y} ⊂ _{A and A is} convex.

2. Considerz∈_{co(A). Then exist}{_λi}k

i=1⊂R+0 and{xi}k_i=1⊂Asatisfying k

X

i=1

λi = 1, z= k

X

i=1

λixi.

SinceAis open, there are{_δi}k

i=1⊂R+ such thatBi :=B(xi, δi)⊂Afor everyi = 1, . . . , k.

Callingδ:= mini=1,...,k{δi}, it is clear that

B(z, δ)⊂

k

X

i=1

λiBi ⊂co(A).

3.LetM∈_{Rsuch that}k_xk ≤_{M. Then choosingy}∈_{co(A), there exist}{_xi}k

i=1inAand

{_λi}k

i=1 satisfyingy=

P

iλixi. Using the triangle inequality we conclude thatkyk ≤M.

4. Letn= dim(X) and consider the map

F: [0,1]n+1×_An+1−→_X given by

F(λ1, . . . , λn+1, x1, . . . , xn+1) =

n+1

X

i=1

λixi

It is clear that

Γ ={_{(λ1, . . . , λn+1)}∈_[0,1]n+1_:

n+1

X

i=1

λi= 1}

is compact, soΓ×_An+1_{is compact in [0,1]}n+1×_An+1_{. Applying theorem1.1,F(Γ}×_An+1_{) =} co(A). SinceFis continuous, co(A) is compact too.

5. GivenA⊂_{X precompact andε}∈_R+_{, there exists a finite set}S ⊂_Asatisfying

A⊂[

x∈S

B(x, ε).

Using the previous result3, co(S_{) is compact and co(A)}⊂ _co(S_{) +B(x, ε) since co(}S_{) +}

B(x, ε) is convex and containsA. Hence it can be found a finite setS₁⊂_co(S_{) such that}

co(S_{) =} [

x∈S₁ B

x,ε

2

.

Now it follows that co(A)⊂ ∪_x∈S

1B(x, ε), showing that co(A) is precompact.

6. The inclusion int(A)⊂_{int(A) is trivial. On the other hand, givenz}∈_{int(A) and}

x ∈_{int(A) with z,x (if z=x is obvious), consider r >0 such thatB(z, r)}⊂_{A and the} point

ω=z+ r

2

z−_x

k_z−_xk ∈B(z, r)⊂A. (1.6) Using lemma1.2is

◦

Sx,ω⊂int(A). Solving the equation1.6forzwe havez=tx+(1−t)ω

wheret=_r+2kr_z−_xk∈]0,1[ andz∈ ◦

Sx,ω⊂int(A).

To prove the other equality, it is clear that int(A)⊂_{A. Reciprocally, givenx}∈_int(A) and z ∈ _{A, is}

◦

Sx,z ⊂ int(A) by lemma 1.2. Hence, taking any sequence {xi}i∈_N ⊂ ◦ Sx,z

which converges toz, we conclude thatz∈_int(A).

Remark 1.1.

Even in finite-dimensional spaces, the application co(·_{) does not always map closed sets to}

closed sets. To give an example,

A=

±_n,1

n

: n∈_N

⊂_R2

is closed butX ₌{_{(x,0) : x}∈_R} ∈_co(A)−_co(A).

-10 -5 5 10

0.2 0.4 0.6 0.8 1.0

Figure 1.4: Illustration ofA.

1.3

Dimension of a convex set

Definition 1.3. LetX be a convex set. For any convex setA⊂_{X, the dimension ofAis the}

dimension of its affine hull:

dim(A) =dim(aff(A)).

There arises now the problem of studying the convex set in its affine hull, in order to get more information. For this circumstance appears the next definition.

Definition 1.4. In a topological vector spaceX, the relative interior of a convex setA⊂_Xis

the interior ofAin the induced topology by its affine hull. The collection of relative interior

points ofAis denoted by ri(A).

It should be appreciated that int(A) and ri(A) are not the same concepts: in fact, givenX=R3andAany unit disk, i.e.,

A={_{(x, y, z)}∈_R3_{: x}2_+y2_{<1, z= 0}}_,

we have thatAis convex, but int(A) =∅_{and ri(A) =A}since aff(_{A) =}{_{(x, y, z)}∈_R3_{: z= 0}}_.

Proposition 1.3. LetX be a topological vector space andAbe a non-empty convex subset

ofX. Then,

1. ri(A),∅.

2. aff(A) =aff(ri(A)). 3. A=ri(A).

Proof. 1. First of all, lemma 1.2 shows that ri(A) is convex. We can suppose without

losing generality that 0∈_{Aand dim(A) =m,0}≤_m≤_{n= dim(X).}

Ifm= 0 it is trivial, so long as A= aff(A) ={₀} _{and ri(A) =}{₀}_{. Otherwise, we can} find{_xi}m

i=1linear-independent vectors that span aff(A) (i.e., forming a basis for aff(A)).

Consider Y =       

x∈_{A: x=}

m

X

i=1

λixi, m

X

i=1

λi <1, λi>0,∀i= 1, . . . , m.

       .

We want to state thatY is open relative to aff(A). To do that, fixy∈_{Y and letx}∈aff(_A). LetMbe then×_{m-matrix which columns are}{_xi}m

i=1andλ, λthe uniquem-dimensional

vectors such that

y=Mλ, x=Mλ.

Due to the fact that MtM is a symmetric and positive definite matrix, we can find

γ∈_R+ _satisfying

k_x−_yk2₌k_M(λ−_λ)k2_{= (M(λ}−_λ))t_(M(λ−_{λ)) = (λ}−_λ)t_Mt_M(λ−_λ)≥_γk_λ−_λk2_. Sincey∈_{Y, the vectorλlies in the open set}

E=       

(λ1, . . . , λm) : m

X

i=1

λi<1, λi>0,∀i = 1, . . . , m

       .

This means thatY contains the intersection of aff(A) and an open ball centred aty, soY is open relative to aff(A). Note that every pointy∈_{Y is a relative interior point of}

A, and hence ri(A),∅.

2. Our previous construction ofY gives us that aff(Y) = aff(A), and sinceY ⊂_ri(A), we see that aff(A) = aff(ri(A)).

3. It is clear that ri(A)⊂_A⇒_ri(A)⊂_{A. On the other hand, lety}∈_Aandx∈_ri(A). Ifx=y, it is done. Otherwise, we know that

◦

Sx,y ⊂ri(A).

Consider the sequence 1

nx+

1−1

n

y

n∈_N

⊂_ri(A).

This sequence converges toy, hencey∈_{ri(A) andA}⊂_ri(A).

A detailed reading of the last proposition gives us an explicit expression of the relative interior of a convex set given by the convex hull of affine-independent points:

ri[co({_{x0, . . . , xk}}_{)] =}       

k

X

i=0

λixi: k

X

i=0

λi= 1, λi >0,∀i = 0, . . . , k

      

.

1.4

Extreme points

We devote the most important section of this chapter to the introduction of the concept of extreme point.

Definition 1.5. An extreme point of a convex set A in a vector space X is a point x ∈ _A

satisfying, for everyy, z∈_A:

x∈_Sy,z⇒_x=y∨_x=z.

We will denote as ext(A)the set of extreme points ofA.

In other words, an extreme point is a point which is not contained in any non-trivial segment of points ofA.

Examples 1.2.

The extreme points of a polyhedron are their own vertexes.

In a closed Euclidean ballA=B(x, r), ext(A) =S(x, r). This example shows that ext(A)

may not be necessarily finite.

Consider the following subset ofX=R3:

A=co({₍±_1,±_1,±₁₎} ∪ {_(cosα,±_{(1 + sinα),0) : a}∈_{[0, π]}}_).

In this case, the points{₍±_1,±_1,0)}_{<ext(A)since they are contained in any segment}

with the formS(±_1,±_1,r),(±_1,±_1,−_r)⊂A,0< r≤1. Now it can be appreciated that ext(A)

Figure 1.5: Plot ofAand co(A).

A more general notion is derived from the previous concept.

Definition 1.6. LetA⊂_{Xa convex set in a vector space. A subsetF}⊂_{Ais said to be a face}

ofAif it is a convex set and, for everyx, y∈_A,

◦

Sx,y∩F,∅ ⇒Sx,y⊂F.

A proper faceF⊂_{AsatisfiesF,A.}

Extreme points are one-point faces of A. A canonical way proper faces are con-structed is via linear functionals.

Proposition 1.4. LetA⊂_{Xa convex set in a vector space andf :A}→_{Ran a}ffi_{ne functional}

withsup_x∈_Af(x) =α <+∞. Then, if

F={_y∈_{A:f(y) =α}} _(1.7)

is a non-empty set, is a face of A. In particular, when X is a topological vector space, any

linear and continuous functional defines a face over a compact convex subset A ⊂_{X in a}

topological vector space.

Proof. It is clear thatF is convex by linearity off. Giveny, z∈_Awith

◦

Sy,z⊂F, we have

        

tf(y) + (1−_{t)f(z) = α}

f(y) ≤ _α

f(z) ≤ _α

⇒_{f(y) =f(z) =α}

andSy,z⊂F.

IfXis a topological vector space and the functional f which appeared in proposi-tion1.4is nonzero, linear, continuous and defined inX, the set given by the equation (1.7) is called an exposed set; in particular, if it is a singleton, we call the point an

Remarks 1.2. LetX be a topological vector space.

In addition to the previous proposition, if the functionalf is nonconstant inA, then

the mentioned face is proper.

Every exposed set F is closed (in the relative topology ofA) by the own definition. In

particular, ifXis HausdorffandA⊂_{X is compact, so isF.}

Every exposed point is an extreme point, but the reciprocal is not true in general. As

an example, considerX=R3 and

A=n(x, y, z)∈_R3_{: x}2_+y2≤_1,−₂≤_z≤_0,o [ n_{(x, y, z)}∈_R3_:x2_+y2_+z2 ≤₁o_.

Every point in the setn(x, y, z)∈_R3_:x2_+y2_{= 1, z= 0}o_{is an extreme one, but it is not}

an exposed one since the unique supporting hyperplane is not a singleton (figure1.6).

Figure 1.6: Example of non-exposed and extreme points.

Recall now the geometric version of Hahn-Banach theorem:

Theorem 1.2(Existence of supporting functionals). LetX be a topological vector space

andAa closed convex subset ofX such that int(A),∅. Then, for anyx0 in the boundary of

A, there exists a nonzero linear and continuous functionalf such that

Ref(x0) = max

x∈_A Ref(x).

Ifα = Ref(x0), the affine hyperplane in XRgiven by H ={x∈X: Ref(x) = α}

con-tainsx0 and isolatesA; we say that the functionalf or the hyperplaneH supports the

setAin the pointx0. One can easily notice that the supporting hyperplane may not be

unique (any vertex of a regular polyhedron admits infinite many of them).

Proposition 1.5. LetXbe a topological vector space andAa convex subset ofX. Any proper

faceF ⊂_{Alies in the boundary ofA. Conversely, ifAis a convex body, then every point of}

its boundary is contained in a proper face.

Proof. Letx∈_Fandy∈_A−_{F. The setB=}{_t∈_{R: tx+ (1}−_t)y⊂_A}_{is contained in [0,1]}

but it can not include anyt >1 for if it did,xwould be an interior point of a segment inAwith at least one point inA−_{F. Hence}

n

(1 +n−1)x+n−1yo

n∈_N

is a sequence inX−_{Awhich converges tox, i.e. x}∈_A∩_X−_A=∂A.

Reciprocally, let us assume that A is a convex body and x0 a point in its

bound-ary. In light of theorem 1.2, there exists a continuous functional f , 0 such that

α = sup_y∈_ARef(y) = Ref(x0). In addition, according to proposition 1.5, the set {x ∈

A: Ref(x) = α} _{defines a proper face of A which contains x0, so long as if Ref was} constant inA, it would be constant inX.

Corollary 1.1. If X is a finite-dimensional topological vector space the dimension of any

proper faceFof a convex setA⊂_{X is strictly less thandim(A).}

Proof. If dim(F) = dim(A), then V = aff(A) = aff(F), hence ri(F), ∅. But F lies in the

boundary ofArelative toV by proposition1.5, so we have a contradiction.

Proposition 1.5highlights the importance of compact sets, so long as it is needed the existence of boundary points (closed sets) and their abundance (bounded sets). Henceforth, we will also restrict the term “face” to indicate a closed set, even if there exist nonclosed faces in infinite dimensional spaces. We conclude this section with an observation about the transitivity of the faces in a convex set.

Proposition 1.6. LetA⊂_{X be a convex set in a vector space andF a face ofA. LetB}⊂_F.

ThenBis a face ofF iffit is a face ofA. In particular,x∈_{Fis in ext(F)i}ff_{it is also in ext(A),}

i.e.,

ext(F) =F∩_ext(A).

Proof. ⇒_{) Suppose thatB} ⊂_{F is a face, x} ∈_{B andx} ∈

◦

Sy,z ⊂A. Since x∈F and F is a

face, we have thaty, z∈_{F. Hencey, z}∈_{Band soBis a face ofA.} ⇐_{) IfB}⊂_{Ais a face,x}∈_Bandx∈

◦

Sy,z⊂F⊂A, then y, z∈F ⊂Aand consequently

y, z∈_{Bfor beingBa face ofA. Thus,Bis a face ofF.}

1.5

Carathéodory-Minkowski theorem

The final section of this chapter will introduce us to Carathéodory-Minkowski the-orem in finite-dimensional spaces. The existence of extreme points will be given by the compactness of the convex set.

Lemma 1.3. LetA⊂_{Xa compact convex set.}

1. Every compact convex setA⊂_{X has at least one extreme point.}

2. Iff :A→_{Ris an a}ffi_{ne functional which attains a unique maximum ina}∈_{A,ais an}

extreme point.

Proof. 1. SinceAis compact andk·k_:X→_{Ris continuous inA, it attains its maximum}

ina∈_{A. Suppose, without losing generality, thata=}1

2(x+y) for someSx,y⊂A. Then,

k_ak2≤1

4(kxk+kyk)

2₌1

4(kxk

2_+kyk2_{+ 2kxk kyk)≤} 1

4(kak

2_+kak2_{+ 2kak kak) =kak}2_.

Thenk_ak₌1

2(kxk+kyk).

Ifk_xk_<k_yk ∨ k_yk ≤ k_xk_{, then}k_ak_<k_yk ∨ k_ak_<k_xk_{, which is a contradiction with} the choice ofa.

Ifk_xk₌k_yk_{, the strict convexity of the Euclidean norm concludes thata=x=y.}

2. It is an easy consequence of proposition1.4.

The final theorem of this chapter is ready to be introduced now.

Theorem 1.3(Carathéodory-Minkowski). LetA⊂_{Xa compact convex subset of a}

finite-dimensional spaceX (with dim(X) =n). Then,

A=co(ext(A)),

namely, everya∈_{Ais a convex combination ofn+ 1extreme points inAas much.}

Proof. It will be done by induction on the dimensionn. Forn= 0,Ais a point and the

result is obvious. Let assume the theorem forn < d. It can be also supposed, without loss of generality that int(A) , ∅. Otherwise we can find an affine variety of lower

dimension (< d) that contains the setAsuch that ri(A) is non-empty (proposition1.3). Since the dimension will be less thand, the result follows from induction hypothesis.

Letaan element in the boundary ofA. By proposition1.5, there exists a faceFsuch thata∈_{F isolatingA. Since dim(F)< d (corollary 1.1) and any face of a compact} con-vex set is a compact concon-vex set, by induction hypothesis is a∈_co(ext(F))⊂_co(ext(A)) (propositions1.1and1.6).

Supposea∈_{int(A). SinceAis bounded, there existx, yin the boundary ofA} satisfy-inga∈

◦

Sx,y ⊂A. As it has been proved,x, y∈co(ext(A)), and since co(ext(A)) is convex,

a∈_co(ext(A)).

After having established Carathéodory-Minkowski’s theorem, it can be fathomed that for every non-empty compact convex subsetA⊂_{X, the set of its extreme points,} ext(A), is always non-empty and it can also be justified the assertion that the convex hull of ext(A), co(ext(A)), is always closed (in fact, compact).

Corollary 1.2. Let A⊂ _{X be a compact convex set and f :A} → _{R a linear (continuous)}

functional. Thenf attains its maximum (or its minimum) at an extreme point ofA.

Proof. By lemma 1.3Ahas an extreme point. Plus, since f is continuous and A

com-pact, f attains its maximum (the other case can be reduced to consider the functional −_{f) in a pointa}∈_{A. Thenais a convex combination of some extreme points ofA, i.e.,}

a=

n+1

X

i=1

λixi, n+1

X

i=1

|_λi|_>0,

n+1

X

i=1

λi = 1,{xi}in=1+1⊂ext(A).

Hence,

f(a) =

n+1

X

i=1

λif(xi)≤ n+1

X

i=1

λif(a) =f(a).

It is clear that, for every λi ,0, f(xi) =f(a), and hence f attains its maximum at an

2

Krein-Milman theorem

Now we have a slight background about finite-dimensional theory, we want to ex-tend those results to an arbitrary dimension (with appropriate considerations). Indeed, we will extend the domain of the space we are going to develop the theory, and we will consider locally convex Hausdorff topological vector spaces; specifically, Krein-Milman theorem states that:

“Every non-empty compact convex subset of a locally convex Hausdorfftopological vector

space is the topological closure of the convex hull of its extreme points”.

This generalisation adds up within our purpose of studying convex sets, since it is only needed the structure of vector space1. In that sense, a first approach to the problem will be given by the algebraic characterisation of unit balls in normed spaces. This step will bring us naturally the main algebraic concepts involved in this theory, which are

balancedness,absorbencyand the own definition of convexity. The immediate

general-isation of those results will make us able to extend the domain to TVS.

The essential tool in the proof will be, as well as in many other results of functional analysis, Zorn’s lemma. The next reminder of the mentioned statement will be useful for the reader:

Theorem 2.1 (Zorn). LetX be a preordered set. If each chain in X has an upper bound,

thenXhas at least one maximal element.

A further relation between the axiom of choice, Zorn’s lemma and Zermelo’s the-orem can be found in [10].

Another useful result will be Hahn-Banach theorem, specifically one form of it which is known as the Geometric Hahn-Banach theorem. It states that:

Theorem 2.2(Hahn-Banach). LetX be a locally convex topological vector space over K=

R∨C. IfA, B are convex, non-empty disjoint subsets of X, Acompact andB closed, then

there exists a continuous linear mapf :X→_{Kands, t}∈_Rsatisfying

Re(f(a))< t < s < Re(f(b)), ∀_a∈_A,∀_b∈_B. In particular,X∗separates points ofX.

To begin with, we will develop some examples of extreme points in infinite-dimensional spaces to show not only the differences between both cases, but also the main strategies that can be developed during the analysis of this theory.

2.1

Examples of extreme points

The most usual examples of topological vector spaces are normed spaces (X,k·k_). Plus, the study of extreme points in some closed ballB(x, r) forx∈_{X andr >0 can be} reduced to the study ofB(0,1). For those reasons, our first examples will be devoted in that environment.

1_{To revise some concepts related to (locally convex) topological vector spaces, (cf. [6]).}