Brouwer fixed point theorem in (L⁰)^d

R E S E A R C H Open Access

Brouwer fixed point theorem in (L ^ ) ^d

Samuel Drapeau¹, Martin Karliczek¹, Michael Kupper^2*and Martin Streckfuß¹

*Correspondence:

kupper@uni-konstanz.de

2Universität Konstanz, Universitätsstraße 10, Konstanz, 78464, Germany

Full list of author information is available at the end of the article

Abstract

The classical Brouwer fixed point theorem states that inR^devery continuous function from a convex, compact set on itself has a fixed point. For an arbitrary probability space, letL⁰=L⁰(,A,P) be the set of random variables. We consider (L⁰)^das an L⁰-module and show that local, sequentially continuous functions onL⁰-convex, closed and bounded subsets have a fixed point which is measurable by construction.

MSC: 47H10; 13C13; 46A19; 60H25

Keywords: conditional simplex; fixed points in (L⁰)^d

Introduction

The Brouwer fixed point theorem states that a continuous function from a compact and convex set inR^dto itself has a fixed point. This result and its extensions play a central role in analysis, optimization and economic theory among others. To show the result, one approach is to consider functions on simplexes first and use Sperner’s lemma.

Recently, Cheriditoet al.[], inspired by the theory developed by Filipovićet al.[] and Guo [], studied (L^)^d as anL^-module, discussing concepts like linear independence, σ-stability, locality andL^-convexity. Based on this, we define affine independence and conditional simplexes in (L^)^d. Showing first a result similar to Sperner’s lemma, we ob- tain a fixed point for local, sequentially continuous functions on conditional simplexes.

From the measurable structure of the problem, it turns out that we have to work with local, measurable labeling functions. To cope with this difficulty and to maintain some uniform properties, we subdivide the conditional simplex barycentrically. We then prove the existence of a measurable completely labeled conditional simplex, contained in the original one, which turns out to be a suitableσ-combination of elements of the barycen- tric subdivision along a partition of. Thus, we can construct a sequence of conditional simplexes converging to a point. By applying always the same rule of labeling using the lo- cality of the function, we show that this point is a fixed point. Due to the measurability of the labeling function, the fixed point is measurable by construction. Hence, even though we follow the constructions and methods used in the proof of the classical result inR^d(cf.

[]), we do not need any measurable selection argument.

In probabilistic analysis theory, the problem of finding random fixed points of random operators is an important issue. GivenC, a compact convex set of a Banach space, a con- tinuous random operator is a functionR:×C→Csatisfying

(i) R(·,x) :→Cis a random variable for any fixedx∈C, (ii) R(ω,·) :C→Cis a continuous function for any fixedω∈.

ForRthere exists a random fixed point which is a random variableξ :→Csuch that ξ(ω) =R(ω,ξ(ω)) for anyω(cf.[–]). In contrast to thisω-wise consideration, our ap-

©2013Drapeau et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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proach is completely within the theory ofL^. All objects and properties are therefore de- fined in that language and proofs are done withL^-methods. Moreover, the connection between continuous random operators onR^dand sequentially continuous functions on (L^)^dis not entirely clear.

An application, though not studied in this paper, is for instance possible in economic theory or optimization in the context of []. Therein the methods from convex analy- sis are used to obtain equilibrium results for translation invariant utility functionals on (L^)^d. Without translation invariance, these methods fail and will be replaced by fixed point arguments in an ongoing work. Thus, our result is helpful to develop the theory of non-translation invariant preference functionals mapping toL^.

The present paper is organized as follows. In the first section, we present the basic con- cepts concerning (L^)^das anL^-module. We define conditional simplexes and examine their basic properties. In the second section, we define measurable labeling functions and show the Brouwer fixed point theorem for conditional simplexes via a construction in the spirit of Sperner’s lemma. In the third section, we show a fixed point result forL^-convex, bounded and sequentially closed sets in (L^)^d. With this result at hand, we present the topological implications known from the real-valued case. On the one hand, we show the impossibility of contracting a ball to a sphere in (L^)^dand, on the other hand, an interme- diate value theorem inL^.

1 Conditional simplex

For a probability space (,A,P), letL^=L^(,A,P) be the space of allA-measurable random variables, whereP-almost surely equal random variables are identified. In partic- ular, forX,Y∈L^, the relationsX≥YandX>Yhave to be understoodP-almost surely.

The setL^with theP-almost everywhere order is a lattice ordered ring, and for a non- empty subsetC⊆L^, we denote the least upper bound byess supCand the greatest lower bound byess infC, respectively (cf.[]). Form∈R, we denote the constant random vari- ablembym. Further, we define the setsL^₊={X∈L^:X≥},L^₊₊={X∈L^:X> } andA+={A∈A:P(A) > }. The set of random variables with values in a setM⊆Ris de- noted byM(A). For example,{, . . . ,r}(A) is the set ofA-measurable functions with values in{, . . . ,r} ⊆N, [, ](A) ={Z∈L^: ≤Z≤}and (, )(A) ={Z∈L^:  <Z< }.

Theconvex hullofX_, . . . ,X_N∈(L^)^d,N∈N, is defined as

conv(X, . . . ,XN) = _N

i=

λiXi:λi∈L^₊, N

i=

λi= 

.

An elementY=N

i=λ_iX_isuch thatλ_i>  for alli∈I⊆ {, . . . ,N}is called astrict convex combination of{Xi:i∈I}. Moreover, a setC⊆(L^)^d is said to beL^-convexif for any X,Y∈Candλ∈[, ](A), it holds thatλX+ ( –λ)Y∈C.

Theσ-stable hullof a setC⊆(L^)^dis defined as σ(C) =

i∈N

_AiX_i:X_i∈C, (Ai)_i∈Nis apartition

,

where a partition is a countable family (A_i)_i∈N⊆Asuch thatP(A_i∩A_j) =  fori=jand P(

i∈NAi) = . We call a non-empty setC σ-stableif it is equal toσ(C). For aσ-stable

setC⊆(L^)^d, a functionf :C→(L^)^dis calledlocaliff(

i∈NAiXi) =

i∈NAif(Xi) for every partition (Ai)i∈N andXi∈C,i∈N. ForX,Y⊆(L^)^d, we call a functionf :X→Y sequentially continuousif for every sequence (X_n)_n∈NinXconverging toX∈XP-almost- surely, it holds thatf(Xn) converges tof(X)P-almost surely. Further, theL^-scalar product andL^-normon (L^)^dare defined as

X,Y= d

i=

XiYi and X= X,X^.

We callC⊆(L^)^dboundedifess sup_X∈CX ∈L^andsequentially closedif it contains all P-almost sure limits of sequences inC. Further, the diameter ofC⊆(L^)^dis defined as diam(C) =ess sup_X,Y∈CX–Y.

Definition . ElementsX, . . . ,XN of (L^)^d,N∈N, are said to beaffinely independent, if eitherN=  orN>  and{Xi–XN}^N–_i= arelinearly independent, that is,

N–

i=

λi(Xi–XN) =  implies λ=· · ·=λN–= , (.)

whereλ_, . . . ,λ_N–∈L^.

The definition of affine independence is equivalent to N

i=

λ_iX_i=  and N

i=

λ_i=  implies λ_=· · ·=λ_N= . (.)

Indeed, first we show that (.) implies (.). Let N

i=λiXi=  andN

i=λi= . Then _N–

i= λ_i(Xi–XN) =λ_NXN +_N–

i= λ_iXi= . By assumption (.), it holds thatλ_=· · ·= λ_N–= , thus alsoλ_N= . To see that (.) implies (.), let_N–

i= λ_i(Xi–XN) = . With λ_N = –_N–

i= λ_i, it holds that_N

i=λ_iX_i=λ_NX_N+_N–

i= λ_iX_i=_N–

i= λ_i(X_i–X_N) = . By assumption (.),λ_=· · ·=λ_N= .

Remark . We observe that if (X_i)^N_i= ⊆(L^)^d are affinely independent, then (λX_i)^N_i=

forλ∈L^₊₊ and (X_i+Y)^N_i= forY ∈(L^)^d are affinely independent. Moreover, if a fam- ilyX, . . . ,XN is affinely independent, then also BX, . . . , BXNare affinely independent on B∈A+, which means from_N

i=Bλ_iXi=  and_N

i=Bλ_i=  it always follows that Bλ_i=  for alli= , . . . ,N.

Definition . Aconditional simplexin (L^)^dis a set of the form S=conv(X, . . . ,X_N)

such thatX, . . . ,XN∈(L^)^dare affinely independent. We callN∈Nthe dimension ofS.

Remark . In a conditional simplexS=conv(X_, . . . ,X_N), the coefficients of convex com- binations are unique in the sense that ifN

i=λi=N

i=μi= , then N

i=

λ_iXi= N

i=

μ_iXi implies λ_i=μ_i for alli= , . . . ,N. (.)

Indeed, since_N

i=(λi–μ_i)Xi=  and_N

i=(λi–μ_i) = , it follows from (.) thatλ_i–μ_i=  for alli= , . . . ,N.

Remark . Note that the present setting -L^-modules and the sequentialP-almost sure convergence - is of local nature. This is, for instance, not the case for subsets ofL^por the convergence in theL^p-norm for ≤p<∞. First,L^p is not closed under multiplication and hence neither a ring nor a module over itself, so that we cannot even speak about affine independence. Second, it is in general not aσ-stable subspace ofL^. However, for a conditional simplexS=conv(X_, . . . ,X_N) in (L^)^dsuch that anyX_k is in (L^p)^d, it holds thatSis uniformly bounded byNsup_k=,...,NXk ∈L^p. This uniform boundedness yields that anyP-almost sure converging sequence inS is also converging in theL^p-norm for

≤p<∞due to the dominated convergence theorem. This shows how one can translate results fromL^toL^p.

Since a conditional simplex is a convex hull, it is in particularσ-stable. In contrast to a simplex inR^d, the representation ofSas a convex hull of affinely independent elements is unique but up toσ-stability.

Proposition . Let(Xi)^N_i=and(Yi)^N_i=be families in(L^)^dwithσ(X, . . . ,XN) =σ(Y, . . . , YN).Thenconv(X, . . . ,XN) =conv(Y, . . . ,YN).Moreover, (Xi)^N_i=are affinely independent if and only if(Y_i)^N_i=are affinely independent.

IfS is a conditional simplex such thatS=conv(X, . . . ,XN) =conv(Y, . . . ,YN),then it holds thatσ(X, . . . ,XN) =σ(Y, . . . ,YN).

Proof Supposeσ(X, . . . ,XN) =σ(Y, . . . ,YN). Fori= , . . . ,N, it holds that Xi∈σ(X, . . . ,XN) =σ(Y, . . . ,YN)⊆conv(Y, . . . ,YN).

Therefore, conv(X, . . . ,XN)⊆ conv(Y, . . . ,YN) and the reverse inclusion holds analo- gously.

Now, let (X_i)^N_i= be affinely independent andσ(X_, . . . ,X_N) =σ(Y_, . . . ,Y_N). We want to show that (Yi)^N_i=are affinely independent. To that end, we define the affine hull

aff(X, . . . ,XN) = _N

i=

λ_iXi:λ_i∈L^, N

i=

λ_i= 

.

First, let Z, . . . ,ZM ∈(L^)^d,M∈N, such thatσ(X, . . . ,XN) =σ(Z, . . . ,ZM). We show that if Aaff(X, . . . ,XN)⊆Aaff(Z, . . . ,ZM) forA∈A+andX, . . . ,XN are affinely inde- pendent, thenM≥N. SinceX_i∈σ(X, . . . ,X_N) =σ(Z, . . . ,Z_M)⊆aff(Z, . . . ,Z_M), we have aff(X, . . . ,XN)⊆aff(Z, . . . ,ZM). Further, it holds thatX=_M

i=_B^

iZifor a partition (B^_i)^M_i=

and hence there exists at least oneB^_ksuch thatA^_k:=B^_k∩A∈A+, and _A^

k

X= _A^

k

Zk. Therefore,

_A^

k

aff(X, . . . ,XN)⊆_A^

k

aff(Z, . . . ,ZM)

= _A^

k

aff {X,Z, . . . ,ZM} \ {Zk} .

ForX=_M

i=_A^

iZi, we find a setA^_ksuch thatA^_k =A^_k∩A^_k∈A+, _A^

k

X= _A^

k

Zk and k_=k_. Assume to the contraryk_=k_, then there exists a setB∈A+such that _BX_=

BX, which is a contradiction to the affine independence of (Xi)^N_i=. Hence, we can again substituteZk byXonA^_k. Inductively, we findk, . . . ,kN such that

A_kNaff(X, . . . ,XN)⊆A_kNaff {X, . . . ,XN,Z, . . . ,ZM} \ {Zk, . . . ,ZkN} ,

which shows M≥N. Now suppose that Y, . . . ,YN are not affinely independent. This means that there exist (λi)^N_i=such that_N

i=λ_iYi=_N

i=λ_i=  but not all coefficientsλ_i are zero, without loss of generality,λ>  onA∈A+. Thus, AY= –AN

i=

λi

λYiand it holds that _Aaff(Y, . . . ,Y_N) = _Aaff(Y, . . . ,Y_N). To see this, consider _AZ= _AN

i=μ_iY_iin

Aaff(Y, . . . ,YN), which means A

_N

i=μ_i= A. Thus, inserting forY,

AZ= A

_N

i=

μ_iYi–μ_ N

i=

λ_i λ_Yi

= A

_N

i=

μ_i–μ_λ_i λ_

Yi

.

Moreover,

A

_N

i=

μ_i–μ_λ_i λ_

= A

_N

i=

μ_i

+ A

–μ_

λ_ N

i=

λ_i

= _A( –μ_) + _Aμ_ λ_λ_= _A.

Hence, it holds that _AZ∈_Aaff(Y_, . . . ,Y_N). It follows that _Aaff(X_, . . . ,X_N) = _Aaff(Y_, . . . , YN) = Aaff(Y, . . . ,YN). This is a contradiction to the former part of the proof (because N– N).

Next, we show that in a conditional simplexS=conv(X, . . . ,X_N) it holds thatXis in σ(X, . . . ,XN) if and only if there do not existY andZinS\ {X}andλ∈(, )(A) such thatλY+ ( –λ)Z=X. ConsiderX∈σ(X_, . . . ,X_N) which isX=N

k=_AkX_kfor a partition (Ak)k=,...,N. Now assume to the contrary that we findY=_N

k=λ_kXkandZ=_N

k=μ_kXkin S\ {X}such thatX=λY+ ( –λ)Z. This means thatX=_N

k=(λλk+ ( –λ)μk)Xk. Due to uniqueness of the coefficients (cf.(.)) in a conditional simplex, we haveλλk+ ( –λ)μk=

Akfor allk= , . . . ,N. By means of  <λ< , it holds thatλλ_k+ ( –λ)μk= Akif and only if λ_k=μ_k= _Ak. Since the last equality holds for allk, it follows thatY=Z=X. Therefore, we cannot findYandZinS\ {X}such thatXis a strict convex combination of them. On the other hand, considerX∈Ssuch thatX∈/σ(X_, . . . ,XN). This meansX=_N

k=ν_kXksuch that there existν_kandν_kandB∈A+with  <ν_k<  onBand  <ν_k<  onB. Defineε:=

ess inf{ν_k,ν_k,  –ν_k,  –ν_k}. Then defineμ_k=λ_k=ν_kifk=k=kandλ_k=ν_k–ε,λ_k= ν_k+ε,μ_k=ν_k+εandμ_k=ν_k–ε. Thus,Y=N

k=λ_kX_kandZ=N

k=μ_kX_kfulfill .Y+

.Z=Xbut both are not equal toXby construction. Hence,Xcan be written as a strict convex combination of elements inS\ {X}. To conclude, considerX∈σ(X_, . . . ,X_N)⊆ S=conv(X, . . . ,XN) =conv(Y, . . . ,YN). SinceX∈σ(X, . . . ,XN), it is not a strict convex combination of elements inS\ {X}, in particular, of elements inconv(Y, . . . ,YN)\ {X}.

Therefore,Xis also inσ(Y, . . . ,YN). Hence,σ(X, . . . ,XN)⊆σ(Y, . . . ,YN). With the same

argumentation, the other inclusion follows.

As an example, let us consider [, ](A). For an arbitraryA∈A, it holds that Aand

A^care affinely independent andconv(A, A^c) ={λA+ ( –λ)A^c: ≤λ≤}= [, ](A).

Thus, the conditional simplex [, ](A) can be written as a convex combination of different affinely independent elements ofL^. This is due to the fact thatσ(, ) ={B:B∈A}= σ(_A, _A^c) for allA∈A.

Remark . In (L^)^d, leteibe the random variable which is  in theith component and  in any other. Then the family ,e, . . . ,edis affinely independent and (L^)^d=aff(,e, . . . ,ed).

Hence, the maximal number of affinely independent elements in (L^)^disd+ .

The characterization ofX∈σ(X, . . . ,XN) leads to the following definition.

Definition . LetS=conv(X_, . . . ,X_N) be a conditional simplex. We define the set of extremal pointsext(S) =σ(X_, . . . ,X_N). For an index setIand a collectionS = (Si)_i∈I of conditional simplexes, we denoteext(S) =σ(

i∈Iext(Si)).

Remark . LetS^j=conv(X_^j, . . . ,X^j_N),j∈N, be conditional simplexes of the same dimen- sionNand (Aj)_j∈Na partition. Then

j∈NAjS^jis again a conditional simplex. To that end, we defineYk=

j∈NAjX^j_kand recognize

j∈NAjS^j=conv(Y, . . . ,YN). Indeed, N

k=

λ_kYk= N

k=

λ_k

j∈N

AjX^j_k=

j∈N

Aj

N k=

λ_kX^j_k∈

j∈N

AjS^j, (.)

showsconv(Y, . . . ,YN)⊆

j∈NAjS^j. The other inclusion follows by considering_N

k=λ^j_k× X_k^j∈S^jand definingλ_k=

j∈N_Ajλ^j_k. To show thatY_, . . . ,Y_Nare affinely independent, we consider_N

k=λ_kYk=  =_N

k=λ_k. Then by (.) it holds that Aj

_N

k=λ_kX_k^j=  and since S^jis a conditional simplex, Ajλ_k=  for allj∈Nandk= , . . . ,N. From the fact that (Aj)_j∈N is a partition, it follows thatλ_k=  for allk= , . . . ,N.

We will prove the Brouwer fixed point theorem in the present setting using an L^- module version of Sperner’s lemma. As in the unconditional case, we have to subdivide a conditional simplex into smaller ones. For our argumentation, we cannot use arbitrary subdivisions and need very special properties of the conditional simplexes in which we subdivide. This leads to the following definition.

Definition . LetS=conv(X, . . . ,XN) be a conditional simplex and SNbe the group of permutations of{, . . . ,N}. Then, forπ∈SN, we define

Y_k^π= k

k i=

Xπ(i), k= , . . . ,N, Cπ=conv Y_^π, . . . ,Y_N^π

.

We call (Cπ)_π∈SN thebarycentric subdivisionofS.

Lemma . Let X, . . . ,XN ∈(L^)^dbe affinely independent.The barycentric subdivision ofS=conv(X, . . . ,XN)is a collection of finitely many conditional simplexes satisfying the following properties:

(i) σ(

π∈SNCπ) =S.

(ii) Cπ has dimensionN,π∈SN.

(iii) Cπ∩Cπis a conditional simplex of dimensionr∈Nandr<Nforπ,π∈S_N,π=π. (iv) Fors= , . . . ,N– ,letBs:=conv(X_, . . . ,X_s).All conditional simplexesCπ∩Bs,

π∈SN,of dimensionssubdivideBsbarycentrically.

Proof We show the affine independence ofY_^π, . . . ,Y_N^π inCπ. It holds that

λ_π()X_π()+λ_π()Xπ()+Xπ()

 +· · ·+λ_{π(N) N}

k=Xπ(k)

N =

N i=

μ_iX_i,

withμ_i=_N

k=π^–(i) λ_π(k)

k . Since_N

i=μ_i=_N

i=λ_i, the affine independence ofY_^π, . . . ,Y_N^π is obtained by the affine independence ofX, . . . ,XN. Therefore allCπ are conditional sim- plexes.

As for Condition (i), it clearly holds thatσ(

π∈SNCπ)⊆S. On the other hand, letX= _N

i=λ_iXi∈S. Then we find a partition (An)n=,...,M, for someM∈N, such that on every Anthe indexes are completely ordered, which isλ_iⁿ

≥λ_iⁿ

 ≥ · · · ≥λ_iⁿ

N onAn.^aThis means thatX∈AnCπⁿwithπⁿ(j) =iⁿ_j. Indeed, we can rewriteXonAnas

X= (λ_iⁿ_–λ_iⁿ

)X_iⁿ_+· · ·+ (N– )(λ_iⁿ

N––λ_iⁿ

N) _N–

k= X_iⁿ

k

N–  +Nλ_iⁿ

N

_N

k=X_iⁿ

k

N ,

which shows thatX∈CπⁿonAn. Condition (ii) is fulfilled by construction.

The intersection of two conditional simplexesCπandCπcan be expressed in the follow- ing manner. LetJ={j:{π(), . . . ,π(j)}={π(), . . . ,π(j)}}be the set of indexes up to which bothπandπhave the same set of images. Then

Cπ∩Cπ=conv Y_j^π:j∈J

. (.)

To show ⊇, let j∈ J. It holds that Y_j^π is in both Cπ and Cπ since {π(), . . . ,π(j)}= {π(), . . . ,π(j)}. Since the intersection ofL^-convex sets isL^-convex, we get this inclu- sion. As for the reverse inclusion, considerX∈Cπ∩Cπ. FromX∈Cπ∩Cπ, it follows that X=_N

i=λ_i(_i

k=

X_π(k) i ) =_N

i=μ_i(_i

k=

X_π(k)

i ). Considerj∈/J. By definition ofJ, there exist p,q≤jwithπ^–(π(p)),π^–(π(q)) /∈ {, . . . ,j}. By (.), the coefficients ofXπ(p) are equal:

N i=p

λ_i i =N

i=π^–(π(p)) μ_i

i. The same holds forX_π(q):N i=q

μ_i i =N

i=π^–(π(q)) λ_i

i. Put together N

i=j+

μ_i i ≤

N i=q

μ_i i =

N i=π^–(π(q))

λ_i i ≤

N i=j+

λ_i i ≤

N i=p

λ_i i =

N i=π^–(π(p))

μ_i i ≤

N i=j+

μ_i i ,

which is only possible ifμ_j=λ_j=  sincep,q≤j. Furthermore, ifCπ∩Cπ is of dimension N, by (.) it follows thatπ=π. This shows (iii).

Further, forBs=conv(X, . . . ,Xs), the elementsCπ ∩Bsof dimensionsare exactly the ones with{π(i) :i= , . . . ,s}={, . . . ,s}. To this end, letCπ ∩Bsbe of dimensions. This means there exists an elementY in this intersection such thatY=N

i=λ_iX_iwithλ_i>  for alli= , . . . ,sandλ_i=  fori>s. As an element ofCπ, thisYhas a representation of the formY=_N

j=(_N

k=j μ_k

k )Xπ(j)for_N

k=μ_k=  andμ_k∈L^₊for everyk= , . . . ,N. Suppose

now that there exists somej≤swithπ(j) >s. Then due toλπ(j)=  and the uniqueness of the coefficients (cf.(.)) in a conditional simplex, it holds that_N

k=j μ_k

k =  and within _N

k=j μ_k

k =  for allj≥j. This meansY=j–

j= (_N

k=j μ_k

k)Xπ(j)and henceY is the convex combination ofj–  elements withj–  <s. This contradicts the property thatλ_i>  for selements. Therefore, (Cπ∩Bs)πis exactly the barycentric subdivision ofBs, which has

been shown to fulfill the properties (i)-(iii).

Subdividing a conditional simplex S = conv(X_, . . . ,XN) barycentrically, we obtain (Cπ)π∈SN. Dividing everyCπbarycentrically results in a new collection of conditional sim- plexes and we call this the two-fold barycentric subdivision ofS. Inductively, we can sub- divide every conditional simplex of the (m– )th step barycentrically and call the resulting collection of conditional simplexes them-fold barycentric subdivision ofSand denote it byS^m. Further, we defineext(S^m) =σ({ext(C) :C∈S^m}) to be theσ-stable hull of all extremal points of the conditional simplexes of them-fold barycentric subdivision ofS.

Notice that this is theσ-stable hull of only finitely many elements, since there are only finitely many simplexes in the subdivision, each of which is the convex hull ofNelements.

Remark . Consider an arbitraryCπ=conv(Y_^π, . . . ,Y_N^π),π∈SNin the barycentric sub- division of a conditional simplexS. Then it holds that

diam(Cπ) =ess sup

i,j=,...,N

Y_i^π–Y_j^π≤N– 

N diam(S).

Since this holds for any π ∈SN, it follows that the diameter ofS^m, which is an arbi- trary conditional simplex of them-fold barycentric subdivision ofS, fulfillsdiam(S^m)≤ (^N–_N )^mdiam(S). Since diam(S) < ∞ and (^N–_N )^m → , for m → ∞, it follows that diam(S^m)→ form→ ∞for every sequence (S^m)_m∈N.

2 Brouwer fixed point theorem for conditional simplexes

Definition . LetS=conv(X_, . . . ,X_N) be a conditional simplex,m-fold barycentrically subdivided inS^m. A local functionφ:ext(S^m)→ {, . . . ,N}(A) is called alabeling func- tionofS. For fixedX, . . . ,XN ∈ext(S) withS=conv(X, . . . ,XN), the labeling function is calledproperif for anyY∈ext(S^m) it holds that

P ω:φ(Y)(ω) =i,λ_i(ω) = 

=  fori= , . . . ,N, whereY=_N

i=λ_iX_i. A conditional simplexC=conv(Y_, . . . ,Y_N), withC⊆ S, withYj∈ext(S^m),j= , . . . ,N, is said to becompletely labeledbyφ if φ is a proper labeling function ofSand

P ω: there existsj∈ {, . . . ,N},φ(Yj)(ω) =i

=  for alli∈ {, . . . ,N}.

Lemma . LetS =conv(X_, . . . ,X_N)be a conditional simplex and f :S →S be a lo- cal function. Let φ :ext(S^m)→ {, . . . ,N}(A) be a local function such that for every X∈ext(S^m)it holds that

(i) P({ω:φ(X)(ω) =i;λ_i(ω) = orμ_i(ω) >λ_i(ω)}) = for alli= , . . . ,N, (ii) P({ω:φ(X)(ω) = ,∃i∈ {, . . . ,N},λi(ω) > ,λi(ω)≥μi(ω)}) = , where(λ_i)_i=,...,Nand(μ_i)_i=,...,Nare determined by X=_N

i=λ_iX_iand f(X) =_N

i=μ_iX_i.Then φis a proper labeling function.

Moreover,the set of functions fulfilling these properties is non-empty.

Proof First we show thatφ is a labeling function. Sinceφis local, we just have to prove thatφactually maps into{, . . . ,N}. Due to (ii), we have to show that

P ω: there existsi∈ {, . . . ,N},λ_i(ω)≥μ_i(ω),λ_i(ω) > 

= .

Assume, to the contrary, thatμ_i>λ_ionA∈A+for allλ_iwithλ_i>  onA. Then it holds that  =_N

i=λ_i_{λi>}<_N

i=μ_i_{μi>}=  onA, which yields a contradiction. Thus,φ is a labeling function. Moreover, due to (i), it holds in particular thatP({ω:φ(X)(ω) =i,λ_i(ω) =

}) = , which shows thatφis proper.

To prove the existence forX∈ext(S^m) withX=_N

i=λ_iXi,f(X) =_N

i=μ_i, letBi:={ω: λi(ω) > } ∩ {ω:λi(ω)≥μi(ω)},i= , . . . ,N. Then we define the functionφ atXas{ω: φ(X)(ω) =i}=B_i\(_i–

k=B_k),i= , . . . ,N. It has been shown thatφmaps to{, . . . ,N}(A) and is proper. It remains to show that φ is local. To this end, considerX=

j∈NAjX^j, where X^j=_N

i=λ^j_iXi andf(X^j) =_N

i=μ^j_iXi. Due to uniqueness of the coefficients in a conditional simplex, it holds thatλ_i=

j∈NAjλ^j_i, and due to locality off, it follows that μ_i=

j∈NAjμ^j_i. Therefore it holds thatBi=

j∈N({ω:λ^j_i(ω) > } ∩ {ω:λ^j_i(ω)≥μ^j_i(ω)} ∩ A_j) =

j∈N(B^j_i∩A_j). Hence,φ(X) =ionB_i\(i–

k=B_k) = [

j∈N(B^j_i∩A_j)]\[i–

k=(

j∈NB^j_k∩ Aj)] =

j∈N[(B^j_i\_i–

k=B^j_k)∩Aj]. On the other hand, we see that

j∈NAjφ(X^j) ision any Aj∩ {ω:φ(X^j)(ω) =i}, hence it isi on

j∈N(B^j_i\_i–

k=B^j_k)∩Aj. Thus,

j∈NAjφ(X^j) = φ(

j∈NAjX^j), which shows thatφis local.

The reason to demand locality of a labeling function is exactly because we want to label by the functionφ mentioned in the existence proof of Lemma . and hence keep local information with it. For example, consider a conditional simplexS=conv(X_,X_,X_,X_) and={ω_,ω_}. LetY∈ext(S) be given byY=_^_

i=Xi. Now consider a functionf on Ssuch that

f(Y)(ω) = 

X(ω) +

X(ω); f(Y)(ω) =

X(ω) +

X(ω) +

X(ω).

If we label Y by the rule explained in Lemma .,φ takes the values φ(Y)(ω)∈ {, }

andφ(Y)(ω) = . Therefore, we can really distinguish on which setsλ_i≥μ_i. Yet, using a deterministic labeling ofY, we would lose this information.

Theorem . LetS=conv(X, . . . ,XN)be a conditional simplex in(L^)^d.Let further f : S →S be a local,sequentially continuous function.Then there exists Y ∈S such that f(Y) =Y.

Proof We consider the barycentric subdivision (Cπ)_π∈SN ofSand a proper labeling func- tionφonext(S). First, we show that we can find a completely labeled conditional simplex inS. By induction on the dimension ofS=conv(X, . . . ,XN), we show that there exists a