Brouwer Fixed Point Theorem in L 0 d

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Brouwer Fixed Point Theorem in L 0 d

Samuel Drapeau

a,1,∗

, Martin Karliczek

a,2,†

, Michael Kupper

b,3,∗

, Martin Streckfuß

a,4,‡

September 12, 2013

ABSTRACT

The classical Brouwer fixed point theorem states that inRd every continuous function from a convex, compact set on itself has a fixed point. For an arbitrary probability space, letL0=L0(Ω,A, P)be the set of random variables. We consider(L0)das anL0-module and show that local, se- quentially continuous functions onL0-convex, closed and bounded subsets have a fixed point which is measurable by construction.

KEYWORDS: Conditional Simplex, Fixed Points in(L0)d, Brouwer

AUTHORSINFO

aHumboldt-Universität Berlin, Unter den Linden 6, 10099 Berlin, Ger- many

bUniversität Konstanz, Universitätsstraße 10, 78464 Konstanz, Germany

1drapeau@math.hu-berlin.de

2karliczm@math.hu-berlin.de

3kupper@uni-konstanz.de

4streckfu@math.hu-berlin.de

∗Funding: MATHEON project E.11

†Funding: Konsul Karl und Dr. Gabriele Sandmann Stiftung

‡Funding: Evangelisches Studienwerk Villigst PAPERINFO ArXiv ePrint:1305.2890

AMS CLASSIFICATION:47H10, 13C13, 46A19, 60H25 We thank Asgar Jamneshan for fruitful discussions.

Introduction

The Brouwer fixed point theorem states that a continuous function from a compact and convex set inRd to 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, Cheridito, Kupper, and Vogelpoth [4], inspired by the theory developed by Filipovi´c, Kup- per, and Vogelpoth [7] and Guo [8], studied (L0)d as an L0-module, discussing concepts like linear independence,σ-stability, locality andL0-convexity. Based on this, we define affine independence and conditional simplexes in(L0)d. Showing first a result similar to Sperner’s Lemma, we obtain 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 barycentric subdivision along a partition ofΩ. Thus, we can construct a sequence of conditional simplexes converging to a point.

By applying always the same rule of labeling using the locality 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 construc- tion. Hence, even though we follow the approach ofRd(cf. [2]) 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 continuous random operator is a functionR: Ω× C → Csatisfying

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

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For R there exists a random fixed point which is a random variable ξ: Ω → C such that ξ(ω) = R(ω, ξ(ω))for anyω(cf. [1], [9], [6]). In contrast to thisω-wise consideration, our approach is com- pletely within the theory ofL0. All objects and properties are therefore defined in that language and proofs are done withL0-methods. Moreover, the connection between continuous random operators on Rdand sequentially continuous functions on(L0)dis not entirely clear.

An application, though not studied in this paper, is for instance possible in economic theory or op- timization in the context of [3]. Therein methods from convex analysis are used to obtain equilibrium results for translation invariant utility functionals on(L0)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 toL0.

The present paper is organized as follows. In the first chapter we present the basic concepts concerning (L0)d as anL0-module. We define conditional simplexes and examine their basic properties. In the second chapter 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 chapter, we show a fixed point result forL0-convex, bounded and sequentially closed sets in(L0)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(L0)dand on the other hand, an intermediate value theorem inL0.

1 Conditional Simplex

For a probability space(Ω,A, P), let L0 = L0(Ω,A, P)be the space of all A-measurable random variables, whereP-almost surely equal random variables are identified. In particular, forX, Y ∈ L0, the relationsX ≥Y andX > Y have to be understoodP-almost surely. The setL0with theP-almost everywhere order is a lattice ordered ring, and every nonempty subsetC ⊆ L0 has a least upper bound ess supC and a greatest lower boundess infC. Form ∈ R, we denote the constant random variable m1Ωbym. Further, we define the setsL0+ = {X ∈ L0:X ≥ 0},L0++ = {X ∈ L0:X > 0}and A+ ={A ∈ A:P(A) >0}. The set of random variables with values in a setM ⊆Ris denoted by M(A). For example,{1, . . . , r}(A)is the set ofA-measurable functions with values in{1, . . . , r} ⊆N, [0,1](A) ={Z∈L0: 0≤Z ≤1}and(0,1)(A) ={Z∈L0: 0< Z <1}.

Theconvex hullofX1, . . . , XN ∈(L0)d,N∈N, is defined as

conv (X1, . . . , XN) = ( N

X

i=1

λiXi:λi∈L0+,

N

X

i=1

λi= 1 )

.

An elementY = PN

i=1λiXi such that λi > 0 for alli ∈ I ⊆ {1, . . . , N}is called astrict convex combinationof{Xi:i∈I}.

Theσ-stable hullof a setC ⊆(L0)dis defined as

σ(C) = (

X

i∈N

1AiXi:Xi∈ C,(Ai)i∈Nis apartition )

,

where a partition is a countable family (Ai)i∈N ⊆ A such that P(Ai ∩Aj) = 0 for i 6= j and P(S

i∈NAi) = 1. We call a nonempty setCσ-stableif it is equal toσ(C). For aσ-stable setC ⊆(L0)da functionf:C →(L0)dis calledlocaliff(P

i∈N1AiXi) =P

i∈N1Aif(Xi)for every partition(Ai)i∈N

andXi ∈ C,i ∈ N. ForX,Y ⊆ (L0)d, we call a function f: X → Ysequentially continuousif for every sequence(Xn)n∈NinX converging toX ∈ X P-almost-surely it holds thatf(Xn)converges to

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f(X)P-almost surely. Further, theL0-scalar productandL0-normon(L0)dare defined as

hX, Yi=

d

X

i=1

XiYi and kXk=hX, Xi12.

We call C ⊆ (L0)d bounded if ess supX∈CkXk ∈ L0 andsequentially closed if it contains allP- almost sure limits of sequences inC. Further, the diameter of C ⊆ (L0)d is defined asdiam(C) = ess supX,Y∈CkX−Yk.

Definition 1.1. ElementsX1, . . . , XN of (L0)d, N ∈ N, are said to be affinely independent, if either N = 1orN >1and{Xi−XN}Ni=1−1arelinearly independent, that is

N−1

X

i=1

λi(Xi−XN) = 0 implies λ1=· · ·=λN−1= 0, (1.1)

whereλ1, . . . , λN−1∈L0.

The definition of affine independence is equivalent to

N

X

i=1

λiXi= 0 and

N

X

i=1

λi= 0 implies λ1=· · ·=λN = 0. (1.2)

Indeed, first, we show that (1.1) implies (1.2). Let PN

i=1λiXi = 0 and PN

i=1λi = 0. Then, PN−1

i=1 λi(Xi−XN) = λNXN +PN−1

i=1 λiXi = 0. By assumption (1.1),λ1 = · · · = λN−1 = 0, thus alsoλN = 0. To see that (1.2) implies (1.1), letPN−1

i=1 λi(Xi−XN) = 0. WithλN =−PN−1 i=1 λi, it holds PN

i=1λiXi = λNXN +PN−1

i=1 λiXi = PN−1

i=1 λi(Xi −XN) = 0. By assumption (1.2), λ1=· · ·=λN = 0.

Remark 1.2. We observe that if(Xi)Ni=1⊆(L0)dare affinely independent then(λXi)Ni=1, forλ∈L0++, and (Xi +Y)Ni=1, for Y ∈ (L0)d, are affinely independent. Moreover, if a family X1, . . . , XN is affinely independent then also1BX1, . . . ,1BXN are affinely independent onB ∈ A+, which means fromPN

i=11BλiXi= 0andPN

i=11Bλi= 0always follows1Bλi= 0for alli= 1, . . . , N. Definition 1.3. Aconditional simplexin(L0)dis a set of the form

S= conv(X1, . . . , XN)

such thatX1, . . . , XN ∈(L0)dare affinely independent. We callN∈Nthe dimension ofS.

Remark 1.4. In a conditional simplexS = conv(X1, . . . , XN), the coefficients of convex combinations are unique in the sense that

N

X

i=1

λiXi=

N

X

i=1

µiXiand

N

X

i=1

λi =

N

X

i=1

µi = 1 implies λi =µifor alli= 1, . . . , N. (1.3)

Indeed, sincePN

i=1(λi−µi)Xi = 0andPN

i=1(λi−µi) = 0, it follows from (1.2) thatλi−µi= 0for

alli= 1, . . . , N.

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Remark 1.5. Note that the present setting —L0-modules and the sequentialP-almost sure convergence

— is of local nature. This is for instance, not the case for subsets ofLp or the convergence in theLp- norm for1≤p <∞. First,Lpis not closed under multiplication and hence neither a ring nor a module over itself so that we can not even speak about affine independence. Second, it is mostly not aσ-stable subspace ofL0. However, for a conditional simplexS= conv(X1, . . . , XN)in(L0)dsuch that anyXkis in(Lp)d, it holds thatSis uniformly bounded byNsupk=1,...,NkXkk ∈Lp. This uniform boundedness yields that anyP-almost sure converging sequence inSis also converging in theLp-norm for1≤p <∞ due to the dominated convergence theorem. This shows how one can translate results fromL0toLp. Since a conditional simplex is a convex hull it is in particularσ-stable. In contrast to a simplex inRd the representation ofSas a convex hull of affinely independent elements is unique but up toσ-stability.

Proposition 1.6. Let(Xi)Ni=1and(Yi)Ni=1be families in(L0)dwithσ(X1, . . . , XN) =σ(Y1, . . . , YN).

Thenconv(X1, . . . , XN) = conv(Y1, . . . , YN). Moreover,(Xi)Ni=1are affinely independent if and only if(Yi)Ni=1are affinely independent.

IfS is a conditional simplex such thatS = conv(X1, . . . , XN) = conv(Y1, . . . , YN), then it holds σ(X1, . . . , XN) =σ(Y1, . . . , YN).

Proof. Supposeσ(X1, . . . , XN) =σ(Y1, . . . , YN). Fori= 1, . . . , N, it holds Xi∈σ(X1, . . . , XN) =σ(Y1, . . . , YN)⊆conv(Y1, . . . , YN).

Therefore,conv(X1, . . . , XN)⊆conv(Y1, . . . , YN)and the reverse inclusion holds analogously.

Now, let(Xi)Ni=1 be affinely independent andσ(X1, . . . , XN) = σ(Y1, . . . , YN). We want to show that(Yi)Ni=1are affinely independent. To that end, we define the affine hull

aff(X1, . . . , XN) = ( N

X

i=1

λiXi:λi∈L0,

N

X

i=1

λi = 1 )

.

First, letZ1, . . . , ZM ∈ (L0)d, M ∈ N, such that σ(X1, . . . , XN) = σ(Z1, . . . , ZM). We show that if1Aaff(X1, . . . , XN)⊆1Aaff(Z1, . . . , ZM)forA∈ A+andX1, . . . , XN are affinely indepen- dent then M ≥ N. SinceXi ∈ σ(X1, . . . , XN) = σ(Z1, . . . , ZM) ⊆ aff(Z1, . . . , ZM), we have aff(X1, . . . , XN)⊆aff(Z1, . . . , ZM). Further, it holds thatX1=PM

i=11B1

iZifor a partition(B1i)Mi=1 and hence there exists at least oneBk1

1 such that A1k

1 := B1k

1 ∩A ∈ A+, and1A1 k1

X1 = 1A1 k1

Zk1. Therefore,

1A1 k1

aff(X1, . . . , XN)⊆1A1 k1

aff(Z1, . . . , ZM) = 1A1 k1

aff({X1, Z1, . . . , ZM} \ {Zk1}).

ForX2 =PM i=11A2

iZiwe find a setA2k, such thatA2k

2 =A2k∩A1k

1 ∈ A+,1A2 k2

X2 = 1A2 k2

Zk2 and k1 6= k2. Assume to the contraryk2 = k1, then there exists a setB ∈ A+, such that1BX1 = 1BX2

which is a contradiction to the affine independence of(Xi)Ni=1. Hence, we can again substituteZk2 by X2onA2k2. Inductively, we findk1, . . . , kN such that

1AkNaff(X1, . . . , XN)⊆1AkN aff({X1, . . . , XN, Z1, . . . , ZM} \ {Zk1, . . . ZkN})

which showsM ≥ N. Now supposeY1, . . . , YN are not affinely independent. This means, there ex- ist(λi)Ni=1 such that PN

i=1λiYi = PN

i=1λi = 0 but not all coefficientsλi are zero, without loss of generality,λ1 > 0 onA ∈ A+. Thus,1AY1 = −1APN

i=2 λi

λ1Yi and it holds1Aaff(Y1, . . . , YN) =

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1Aaff(Y2, . . . , YN). To see this, consider 1AZ = 1APN

i=1µiYi ∈ 1Aaff(Y1, . . . , YN)which means 1APN

i=1µi= 1A. Thus, inserting forY1,

1AZ= 1A

"N X

i=2

µiYi−µ1 N

X

i=2

λi

λ1

Yi

#

= 1A

"N X

i=2

µi−µ1

λi

λ1

Yi

# .

Moreover,

1A

" N X

i=2

µi−µ1

λi

λ1 #

= 1A

" N X

i=2

µi

# + 1A

"

−µ1

λ1

N

X

i=2

λi

#

= 1A(1−µ1) + 1A

µ1

λ1λ1= 1A. Hence, 1AZ ∈ 1Aaff(Y2, . . . , YN). It follows that 1Aaff(X1, . . . , XN) = 1Aaff(Y1, . . . , YN) = 1Aaff(Y2, . . . , YN). This is a contradiction to the former part of the proof (becauseN−16≥N).

Next, we characterize extremal points inS = conv(X1, . . . , XN). To this end, we show X ∈ σ(X1, . . . , XN)if and only if there do not exist Y and Z in S \ {X} andλ ∈ (0,1)(A) such that λY + (1−λ)Z = X. ConsiderX ∈ σ(X1, . . . , XN)which isX = PN

k=11AkXk for a partition (Ak)k∈N. Now assume to the contrary that we findY =PN

k=1λkXkandZ=PN

k=1µkXkinS \ {X} such thatX=λY + (1−λ)Z. This means thatX =PN

k=1(λλk+ (1−λ)µk)Xk. Due to uniqueness of the coefficients (cf. (1.3)) in a conditional simplex we haveλλk+ (1−λ)µk = 1Akfor allk= 1. . . , N.

By means of0 < λ < 1, it holds thatλλk + (1−λ)µk = 1Ak if and onlyλk = µk = 1Ak. Since the last equality holds for allk it follows thatY = Z = X. Therefore, we cannot find Y andZ in S \ {X}such thatX is a strict convex combination of them. On the other hand, considerX ∈ S such thatX /∈σ(X1, . . . , XN). This means,X =PN

k=1νkXk, such that there existνk1andνk2andB∈ A+

with0 < νk1 < 1on B and0 < νk2 < 1 onB. Define ε := ess inf{νk1, νk2,1−νk1,1−νk2}.

Then defineµk =λk = νk ifk1 6=k 6=k2andλk1 = νk1 −ε,λk2 = νk2 +ε,µk1 = νk1 +εand µk2 =νk2−ε. Thus,Y =PN

k=1λkXk andZ =PN

k=1µkXkfulfill0.5Y + 0.5Z =X but both are not equal toX by construction. Hence,Xcan be written as a strict convex combination of elements in S \ {X}. To conclude, considerX∈σ(X1, . . . , XN)⊆ S= conv(X1, . . . , XN) = conv(Y1, . . . , YN).

SinceX∈σ(X1, . . . , XN)it is not a strict convex combinations of elements inS \ {X}, in particular, of elements inconv(Y1, . . . , YN)\ {X}. Therefore,Xis also inσ(Y1, . . . , YN). Hence,σ(X1, . . . , XN)⊆ σ(Y1, . . . , YN). With the same argumentation the other inclusion follows.

As an example let us consider[0,1](A). For an arbitraryA∈ A, it holds that1Aand1Acare affinely independent andconv(1A,1Ac) ={λ1A+ (1−λ)1Ac: 0≤λ≤1}= [0,1](A). Thus, the conditional simplex[0,1](A)can be written as a convex combination of different affinely independent elements of L0. This is due to the fact thatσ(0,1) ={1B :B∈ A}=σ(1A,1Ac)for allA∈ A.

Remark 1.7. In(L0)d, leteibe the random variable which is1in thei-th component and0in any other.

Then the family0, e1, . . . , edis affinely independent and(L0)d= aff(0, e1, . . . , ed). Hence, the maximal

number of affinely independent elements in(L0)disd+ 1.

The characterization ofX ∈σ(X1, . . . , XN)leads to the following definition.

Definition 1.8. LetS = conv(X1, . . . , XN)be a conditional simplex. We define the set of extremal pointsext(S) = σ(X1, . . . , XN). For an index set I and a collection S = (Si)i∈I of conditional simplexes we denoteext(S) =σ(ext(Si) :i∈I).

Remark 1.9. LetSj = conv(X1j, . . . , XNj),j ∈ N, be conditional simplexes of the same dimension N and(Aj)j∈Na partition. ThenP

j∈N1AjSj is again a conditional simplex. To that end, we define

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Yk=P

j∈N1AjXkjand recognizeP

j∈N1AjSj= conv(Y1, . . . , YN). Indeed,

N

X

k=1

λkYk=

N

X

k=1

λkX

j∈N

1AjXkj =X

j∈N

1Aj

N

X

k=1

λkXkj∈X

j∈N

1AjSj, (1.4)

showsconv(Y1, . . . , YN) ⊆P

j∈N1AjSj. The other inclusion follows by consideringPN

k=1λjkXkj ∈ Sj and definingλk = P

j∈N1Ajλjk. To show thatY1, . . . , YN are affinely independent, we consider PN

k=1λkYk = 0 =PN

k=1λk. Then by (1.4), it holds1AjPN

k=1λkXkj = 0and sinceSjis a conditional simplex,1Ajλk = 0for allj∈Nandk= 1, . . . , N. From the fact that(Aj)j∈Nis a partition, it follows

thatλk= 0for allk= 1, . . . , N.

We will prove the Brouwer fixed point theorem in the present setting using an analogue version of Sperner’s lemma. As in the unconditional case we have to subdivide a conditional simplex in 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 1.10. LetS = conv(X1, . . . , XN)be a conditional simplex andSNthe group of permutations of{1, . . . , N}. Then forπ∈SNwe define

Cπ= conv

Xπ(1),Xπ(1)+Xπ(2)

2 , . . . ,Xπ(1)+· · ·+Xπ(k)

k , . . . ,Xπ(1)+· · ·+Xπ(N) N

.

We call(Cπ)π∈SN thebarycentric subdivisionofS, and denoteYkπ= 1kPk

i=1Xπ(i).

Lemma 1.11. The barycentric subdivision is a collection of finitely many conditional simplexes satisfying the following properties

(i) σ(S

π∈SNCπ) =S.

(ii) Cπhas dimensionN,π∈SN.

(iii) Cπ∩ Cπis a conditional simplex of dimensionr∈Nandr < N forπ, π∈SN,π6=π.

(iv) Fors= 1, . . . , N−1, letBs:= conv(X1, . . . , Xs). All conditional simplexesCπ∩ Bs,π∈SN, of dimensionssubdivideBsbarycentrically.

Proof. We show the affine independence ofY1π, . . . , YNπinCπ. It holds

λπ(1)Xπ(1)+λπ(2)Xπ(1)+Xπ(2)

2 +· · ·+λπ(N) PN

k=1Xπ(k)

N =

N

X

i=1

µiXi, withµi = PN

k=π−1(i) λπ(k)

k . Since PN

i=1µi = PN

i=1λi, the affine independence ofY1π, . . . , YNπ is obtained by the affine independence ofX1, . . . , XN. Therefore allCπare conditional simplexes.

The intersection of two conditional simplexesCπandCπcan be expressed in the following manner. Let J ={j:{π(1), . . . , π(j)}={π(1), . . . , π(j)}}be the set of indexes up to which bothπandπhave the same set of images. Then,

Cπ∩ Cπ= conv Pj

k=1Xπ(k) j :j∈J

!

. (1.5)

To show ⊇, let j ∈ J. It holds that

Pj k=1Xπ(k)

j is in both Cπ andCπ since{π(1), . . . , π(j)} = {π(1), . . . , π(j)}. Since the intersection of convex sets is convex, we get this implication.

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As for the reverse inclusion, consider X ∈ Cπ∩ Cπ. FromX ∈ Cπ ∩C¯π, it follows thatX = PN

i=1λi(Pi k=1

Xπ(k)

i ) =PN

i=1µi(Pi k=1

Xπ(k)

i ). Considerj6∈J. By definition ofJ, there existp, q≤ jwithπ−1(π(p)), π−1(π(q)) 6∈ {1, . . . , j}. By (1.3), the coefficients ofXπ(p)are equal: PN

i=p λi

i = PN

i=π−1(π(p)) µi

i. The same holds forXπ(q):PN i=q

µi i =PN

i=π−1(π(q)) λi

i . Put together

N

X

i=j+1

µi

i ≤

N

X

i=q

µi

i =

N

X

i=π−1(π(q))

λi

i ≤

N

X

i=j+1

λi

i ≤

N

X

i=p

λi

i =

N

X

i=π−1(π(p))

µi

i ≤

N

X

i=j+1

µi

i which is only possible ifµj=λj= 0sincep, q≤j.

Furthermore, ifCπ∩ Cπis of dimensionNby (1.5) follows thatπ=π. This shows(iii).

As for Condition(i), it clearly holdsσ(∪π∈SNCπ)⊆ S. On the other hand, letX =PN

i=1λiXi ∈ S.

Then, we find a partition(An)n∈Nsuch that on everyAn the indexes are completely ordered which is λin1 ≥λin2 ≥ · · · ≥λinN onAn. This means, thatX ∈1AnCπnwithπn(j) =inj. Indeed, we can rewrite XonAnas

X = (λin1 −λin2)Xin1 +· · ·+ (N−1)(λinN−1−λinN) PN−1

k=1 Xin

k

N−1 +N λinN

PN k=1Xin

k

N ,

which shows thatX ∈ CπnonAn.

Further, forBs= conv(X1, . . . , Xs)the elementsCπ0∩ Bsof dimensionsare exactly the ones with {π0(i) :i= 1, . . . , s}={1, . . . , s}. To this end, letCπ0∩ Bsbe of dimensions. This means there exists an elementY in this intersection such thatY =PN

i=1λiXiwithλi>0for alli= 1, . . . , sandλi= 0 fori > s. As an element ofCπ0 thisY has a representation of the formY =PN

j=1(PN k=j

µk

k )Xπ0(j), forPN

k=1µk= 1andµk ∈L0+for everyk= 1, . . . , N. Suppose now that there exists somej0≤swith π0(j0)> s. Then due toλπ0(j0)= 0and the uniqueness of the coefficients (cf.(1.3)) in anL0-simplex, it holdsPN

k=j0 µk

k = 0and withinPN k=j

µk

k = 0for allj ≥j0. This meansY =Pj0−1 j=1 (PN

k=j µk

k )Xπ0(j)

and henceY is the convex combination ofj0−1elements withj0−1< s. This contradicts the property thatλi >0forselements. Therefore,(Cπ0 ∩ Bs)π0 is exactly the barycentric subdivision ofBs, which

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

Subdividing a conditional simplexS = conv(X1, . . . , XN)barycentrically we obtain(Cπ)π∈SN. Di- viding everyCπ barycentrically results in a new collection of conditional simplexes and we call this the two-fold barycentric subdivision ofS. Inductively, we can subdivide every conditional simplex of the(m−1)th step barycentrically and call the resulting collection of conditional simplexes them-fold barycentric subdivision ofS and denote it bySm. Further, we define ext(Sm) = σ({ext(C) : C ∈ Sm})to be theσ-stable hull of all extremal points of the conditional simplexes of them-fold barycen- tric 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 1.12. Consider an arbitraryCπ = conv(Y1π, . . . , YNπ),π∈SNin the barycentric subdivision of a conditional simplexS. Then it holds

diam(Cπ)≤ ess sup

i,j=1,...,N

Yiπ−Yjπ

≤N−1

N diam(S).

Since this holds for anyπ ∈ SN, it follows that the diameter ofSm, which is an arbitrary conditional simplex of the m-fold barycentric subdivision ofS, fulfillsdiam(Sm) ≤ N−1N m

diam(S). Since diam(S)<∞and NN−1m

→0, form→ ∞, it follows thatdiam(Sm)→0, form→ ∞for every

sequence(Sm)m∈N.

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2 Brouwer Fixed Point Theorem for Conditional Simplexes

Definition 2.1. LetS= conv(X1, . . . , XN)be a conditional simplex,m-fold barycentrically subdivided inSm. A local functionφ: ext(Sm)→ {1, . . . , N}(A)is called alabeling functionofS. For fixed X1, . . . , XN ∈ext(S)withS = conv(X1, . . . , XN), the labeling function is calledproper, if for any Y ∈ext(Sm)it holds that

P({φ(Y) =i} ⊆ {λi>0}) = 1, fori = 1, . . . , N, whereY = PN

i=1λiXi. A conditional simplexC = conv(Y1, . . . , YN) ⊆ S, with Yj ∈ext(Sm), j= 1, . . . , N, is said to becompletely labeledbyφif this is a proper labeling function ofSand

P

N

[

j=1

{φ(Yj) =i}

= 1,

for alli∈ {1, . . . , N}.

Lemma 2.2. LetS = conv(X1, . . . , XN)be a conditional simplex andf:S → S be a local function.

Letφ: ext(Sm)→ {0, . . . , N}(A)be a local function such that

(i) P({φ(X) =i} ⊆ {λi >0} ∩ {λi≥µi}) = 1, for alli= 1, . . . , N, (ii) P

SN

i=1 {λi >0} ∩ {λi≥µi}

⊆SN

i=1{φ(X) =i}

= 1, whereX=PN

i=1λiXiandf(X) =PN

i=1µiXi. Then,φis a proper labeling function.

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

Proof. First we show thatφis a labeling function. Sinceφis local we just have to prove thatφactually maps to{1, . . . , N}. Due to(ii), we have to show thatP(SN

i=1{λi≥µi:λi >0}) = 1. Assume to the contrary,µi > λi onA∈ A+, for allλiwithλi >0onA. Then it holds that1 = PN

i=1λi1{λi>0} <

PN

i=1µi1{µi>0} = 1onAwhich yields a contradiction. Thus,φis a labeling function. Moreover, due to(i)it holds thatP({φ(X) =i} ⊆ {λi>0}) = 1which shows thatφis proper.

To prove the existence, for X ∈ ext(Sm)withX = PN

i=1λiXi, f(X) = PN

i=1µi let Bi :=

{λi > 0} ∩ {λi ≥ µi}, i = 1, . . . , N. Then we define the function φ at X as {φ(X) = i} = Bi \(Si−1

k=1Bk),i = 1, . . . , N. It has been shown thatφmaps to{1, . . . , N}(A) and is proper. It remains to show that φis local. To this end, considerX = P

j∈N1AjXj whereXj = PN i=1λjiXi and f(Xj) = PN

i=1µjiXi. Due to uniqueness of the coefficients in a conditional simplex it holds that λi = P

j∈N1Ajλji and due to locality of f it follows that µi = P

j∈N1Ajµji. Therefore it holds that Bi = S

j∈N

{λji >0} ∩ {λji ≥µji} ∩Aj

= S

j∈N(Bji ∩Aj). Hence, φ(X) = i on Bi \ (Si−1

k=1Bk) = [S

j∈N(Bji ∩Aj)]\[Si−1 k=1(S

j∈NBkj ∩Aj)] = S

j∈N[(Bij \Si−1

k=1Bkj)∩ Aj].

On the other hand, we see that P

j∈N1Ajφ(Xj) is i on any Aj ∩ {φ(Xj) = i}, hence it is i on S

j∈N(Bij\Si−1

k=1Bkj)∩Aj. Thus,P

j∈N1Ajφ(Xj) =φ(P

j∈N1AjXj)which shows thatφis local.

The reason to demand locality of a labeling function is exactly because we want to label by the rule explained in Lemma2.2and hence keep local information with it. For example consider a conditional simplexS= conv(X1, X2, X3, X4)andΩ ={ω1, ω2}. LetY ∈ext(S)be given byY =13P3

i=1Xi. Now consider a functionfonSsuch that

f(Y)(ω1) = 1

4X1(ω1) +3

4X3(ω1); f(Y)(ω2) =2

5X1(ω2) +2

5X2(ω2) +1

5X4(ω2).

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If we labelY by the rule explained in Lemma2.2,φtakes the valuesφ(Y)(ω1)∈ {1,2}andφ(Y)(ω2) = 3. Therefore, we can really distinguish on which setsλi ≥µi. Yet, using a deterministic labeling ofY, we would loose this information.

Theorem 2.3. LetS= conv(X1, . . . , XN)be a conditional simplex in(L0)d. Letf:S → Sbe a local, sequentially continuous function. Then there existsY ∈ Ssuch thatf(Y) =Y.

Proof. We consider the barycentric subdivision (Cπ)π∈SN of S and a proper labeling function φ on ext(S). First, we show that we can find a completely labeled conditional simplex inS. By induc- tion on the dimension ofS = conv(X1, . . . , XN), we show that there exists a partition(Ak)k=1,...,K

such that on anyAk there is an odd number of completely labeledCπ. The caseN = 1is clear, since a point can be labeled with the constant index1, only.

Suppose the caseN −1 is proven. Since the number ofYiπ of the barycentric subdivision is finite andφcan only take finitely many values, it holds for allV ∈(Yiπ)i=1,...,N,π∈SN there exists a partition (AVk)k=1,...,K,K <∞, whereφ(V)is constant on anyAVk. Therefore, we find a partition(Ak)k=1,...,K, such thatφ(V)onAkis constant for allV andAk. FixAk now.

In the following, we denote byCπb these conditional simplexes for whichCπb∩ BN−1 are N −1- dimensional (cf. Lemma1.11 (iv)), thereforeπb(N) =N. Further we denote byCπc these conditional simplexes which are not of the typeCπb, that isπc(N)6=N. If we useCπwe mean a conditional simplex of arbitrary type. We define

• C ⊆(Cπ)π∈SN to be the set ofCπwhich are completely labeled onAk.

• A ⊆(Cπ)π∈SN to be the set of theP-almost completely labeledCπ, that is {φ(Ykπ), k= 1, . . . , N}={1, . . . , N −1} onAk.

• Eπto be the set of the intersections(Cπ∩ Cπl)πl∈SNwhich areN−1-dimensional and completely labeled onAk.1

• Bπto be the set of the intersectionsCπ∩ BN−1which are completely labeled onAk.

It holds thatEπ∩Bπ =∅and hence|Eπ∪Bπ|=|Eπ|+|Bπ|. SinceCπc∩ BN−1is at mostN−2- dimensional, it holds thatBπc =∅and hence|Bπc|= 0. Moreover, we know thatCπ∩ Cπl isN−1- dimensional onAk if and only if this holds on wholeΩ(cf. Lemma1.11 (iii)) andCπb∩ BN−1 6=∅on Ak if and only if this also holds on wholeΩ(cf. Lemma1.11 (iv)). So it does not play any role if we look at these sets which are intersections onAkor onΩsince they are exactly the same sets.

IfCπc ∈C then|Eπc| = 1and ifCπb ∈C then|Eπb∪Bπb|= 1. IfCπc ∈A then|Eπc|= 2and if Cπb ∈A then|Eπb∪Bπb|= 2. Therefore it holdsP

π∈SN|Eπ∪Bπ|=|C|+ 2|A|.

If we pick anEπ∈Eπwe know there always exists exactly one otherπlsuch thatEπ ∈Eπl(Lemma 1.11(iii)). ThereforeP

π∈SN|Eπ|is even. Moreover(Cπb∩ BN−1)πbsubdividesBN−1barycentrically2 and hence we can apply the hypothesis (onext(Cπb∩ BN−1)). This means that the number of completely labeled conditional simplexes is odd on a partition ofΩbut sinceφis constant onAkit also has to be odd there. This means thatP

πb|Bπb|has to be odd. Hence, we also have thatP

π|Eπ∪Bπ|is the sum of an even and an odd number and thus odd. So we conclude|C|+ 2|A|is odd and hence also|C|. Thus, we find for anyAka completely labeledCπk.

1That is bearing exactly the label1, . . . , N−1onAk.

2 The boundary ofS is aσ-stable set so if it is partitioned by the labeling function intoAkwe know thatBN−1(S) = PK

k=11AkBN−1(1AkS)and by Lemma1.11 (iv)we can apply the induction hypothesis also onAk.

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We defineS1 = PK

k=11AkCπk which by Remark 1.9is indeed a conditional simplex. Due toσ- stability ofSit holdsS1 ⊆ S. By Remark1.12S1has a diameter which is less then NN−1diam(S)and sinceφis localS1is completely labeled on wholeΩ.

The same argumentation holds for everym-fold barycentric subdivisionSm ofS,m ∈ N, that is, there exists a completely labeled conditional simplex in every m-fold barycentrically subdivided con- ditional simplex which is properly labeled. Henceforth, subdividing S m-fold barycentrically and la- bel it byφm: ext(Sm) → {1, . . . , N}(A), which is a labeling function as in Lemma 2.2, we al- ways obtain a completely labeled conditional simplex Sm+1 ⊆ S, form ∈ N. Moreover, sinceS1 is completely labeled, it holdsS1 = PK

k=11AkCπk as above whereCπk is completely labeled on Ak. This means Cπk = conv(Y1k, . . . , YNk) with φ(Yjk) = j on Ak for every j = 1, . . . , N. Defining Vj1 = PK

k=11AkYjk for everyj = 1, . . . , N yieldsP({φ(Vj1) = j}) = 1for every j = 1, . . . , N and S1 = conv(V11, . . . , VN1). The same holds for any m ∈ N and so that we can write Sm = conv(V1m, . . . , VNm)withP(φm−1({Vjm) =j}) = 1for everyj= 1, . . . , N.

Now,(V1m)m∈Nis a sequence in the sequentially closed,L0-bounded setS, so that by [4, Corollary 3.9], there existsY ∈ S and a sequence(Mm)m∈N inN(A)such thatMm+1 > Mm for allm ∈ N andlimm→∞V1Mm = Y P-almost surely. ForMm ∈ N(A),V1Mm is defined asP

n∈N1{Mm=n}V1n. This means an element with index Mm, for some m ∈ N, equals V1n onAn,n ∈ N, where the sets An are determined byMmviaAn = {Mm =n},n∈ N. Furthermore, asmgoes to∞,diam(Sm) is converging to zeroP-almost surely, and therefore it also follows thatlimm→∞VkMm =Y P-almost surely for everyk = 1, . . . , N. Indeed, it holds|Vkm−Y| ≤ diam(Sm) +|V1m−Y|for everyk = 1, . . . , N andm∈Nso we can use the sequence(Mm)m∈Nfor everyk= 1, . . . , N.

Let Y = PN

l=1αlXl and f(Y) = PN

l=1βlXl as well as Vkm = PN

l=1λm,kl Xl and f(Vkm) = PN

l=1µm,kl Xl for m ∈ N. As f is local it holds that f(V1Mm) = P

n∈N1{Mm=n}f(V1n). By se- quential continuity of f, it follows that limm→∞f(VkMn) = f(Y)P-almost surely for every k = 1, . . . , N. In particular, limm→∞λMl m,l = αl andlimm→∞µMl m,l = βl P-almost surely for every l = 1, . . . , N. However, by construction,φm−1(Vlm) =lfor everyl = 1, . . . , N, and from the choice of φm−1, it follows that λm,ll ≥ µm,ll P-almost surely for every l = 1, . . . , N andm ∈ N. Hence, αl= limm→∞λMl m,l≥limm→∞µMl m,l =βlP-almost surely for everyl= 1, . . . , N. This is possible only ifαl=βlP-almost surely for everyl= 1, . . . , N, showing thatf(Y) =Y.

3 Applications

3.1 Fixed point theorem for sequentially closed and bounded sets in (L

0

)

d

Proposition 3.1. LetKbe anL0-convex, sequentially closed and bounded subset of(L0)dandf:K → Ka local, sequentially continuous function. Thenf has a fixed point.

Proof. Since K is bounded, there exists a conditional simplex S such that K ⊆ S. Now define the functionh:S → Kby

h(X) =

(X, ifX ∈ K, arg min{kX−Yk:Y ∈ K}, else.

This means, thathis the identity onKand the projection onKfor the elements inS \ K. Due to [4, Corollary 5.5] this minmium exists and is unique. Thereforehis well-defined.

We can characterizehby

Y =h(X)⇔ hX−Y, Z−Yi ≤0, for allZ∈ K. (3.1)

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Indeed, lethX−Y, Z−Yi ≤0for allZ∈ K. Then kX−Zk2=k(X−Y) + (Y −Z)k

=kX−Yk2+ 2hX−Y, Y −Zi+kY −Zk2≥ kX−Yk2, which shows the minimizing property of h. On the other hand, let Y = h(X). Since K is convex, λZ+ (1−λ)Y ∈ Kfor anyλ∈(0,1](A)andZ ∈ K. By standard calculation,

kX−(λZ+ (1−λ)Y)k2≥ kX−Yk2

yields0 ≥ −2λhX,−Yi+ (2λ−λ2)hY, Yi+ 2λhX, Zi −λ2kZk2−2λ(1−λ)hZ, Yi. Dividing by λ >0and lettingλ↓0afterwards yields

0≥ −2hX,−Yi+ 2hY, Yi+ 2hX, Zi −2hZ, Yi= 2hX−Y, Z−Yi, which is the desired claim.

Furthermore, for anyX, Y ∈ Sholds

kh(X)−h(Y)k ≤ kX−Yk. Indeed,

X−Y = (h(X)−h(Y)) +X−h(X) +h(Y)−Y =: (h(X)−h(Y)) +c which means

kX−Yk2=kh(X)−h(Y)k2+kck2+ 2hc, h(X)−h(Y)i. (3.2) Since

hc, h(X)−h(Y)i=−hX−h(X), h(Y)−h(X)i − hY −h(Y), h(X)−h(Y)i,

by (3.1), it follows thathc, h(X)−h(Y)i ≥ 0and (3.2) yieldskX−Yk2 ≥ kh(X)−h(Y)k2. This shows thathis sequentially continuous.

The functionf◦his a sequentially continuous function mapping fromStoK ⊆ S. Hence, there exists a fixed pointf ◦h(Z) =Z. Sincef ◦hmaps toK, thisZhas to be inK. But then we knowh(Z) =Z

and thereforef(Z) =Zwhich ends the proof.

Remark 3.2. In Drapeau, Jamneshan, Karliczek, and Kupper [5] the concept of conditional compactness is introduced and it is shown that there is an equivalence between conditional compactness and conditional closed- and boundedness in(L0)d. In that context we can formulate the conditional Brouwer fixed point theorem as follows. A sequentially continuous functionf:K → Ksuch thatKis a conditionally compact

andL0-convex subset of(L0)dhas a fixed point.

3.2 Applications in Conditional Analysis on (L

0

)

d

Working inRdthe Brouwer fixed point theorem can be used to prove several topological properties and is even equivalent to some of them. In the theory of(L0)dwe will shown that the conditional Brouwer fixed point theorem has several implications as well.

Define theunit ballin(L0)dbyB(d) ={X ∈(L0)d: kXk ≤ 1}. Then by the former theorem any local, sequentially continuous functionf:B(d)→ B(d)has a fixed point. Theunit sphereS(d−1)is defined asS(d−1) ={X∈(L0)d: kXk= 1}.

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