R E S E A R C H
Open Access
Brouwer fixed point theorem in
(
L
)
d
Samuel Drapeau
1, Martin Karliczek
1, Michael Kupper
2*and Martin Streckfuß
1*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 inRdevery 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 inRdto 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 inRd(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
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 onRdand 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 setLwith 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-ablembym. 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, . . . ,XN∈(L)d,N∈N, is defined as
conv(X, . . . ,XN) =
N
i=
λiXi:λi∈L+,
N
i= λi=
.
An elementY=Ni=λiXisuch thatλi> 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
AiXi:Xi∈C, (Ai)i∈Nis apartition
,
where a partition is a countable family (Ai)i∈N⊆Asuch thatP(Ai∩Aj) = fori=jand
setC⊆(L)d, a functionf :C→(L)dis calledlocaliff(
i∈NAiXi) =
i∈NAif(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 (Xn)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 ∈Landsequentially closedif it contains all
P-almost sure limits of sequences inC. Further, the diameter ofC⊆(L)dis defined as
diam(C) =ess supX,Y∈CX–Y.
Definition . ElementsX, . . . ,XN of (L)d,N∈N, are said to beaffinely independent, if eitherN= orN> and{Xi–XN}Ni=–arelinearly independent, that is,
N–
i=
λi(Xi–XN) = implies λ=· · ·=λN–= , (.)
whereλ, . . . ,λN–∈L.
The definition of affine independence is equivalent to
N
i=
λiXi= and N
i=
λi= implies λ=· · ·=λN= . (.)
Indeed, first we show that (.) implies (.). Let Ni=λiXi= andNi=λi= . Then
N–
i= λi(Xi–XN) =λNXN +
N–
i= λiXi= . By assumption (.), it holds thatλ=· · ·= λN–= , thus alsoλN= . To see that (.) implies (.), let
N–
i= λi(Xi–XN) = . With
λN = –
N–
i= λi, it holds that
N
i=λiXi=λNXN+
N–
i= λiXi=
N–
i= λi(Xi–XN) = . By assumption (.),λ=· · ·=λN= .
Remark . We observe that if (Xi)Ni= ⊆(L)d are affinely independent, then (λXi)Ni=
forλ∈L
++ and (Xi+Y)Ni= 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=BλiXi= and
N
i=Bλi= it always follows that Bλi= for alli= , . . . ,N.
Definition . Aconditional simplexin (L)dis a set of the form
S=conv(X, . . . ,XN)
such thatX, . . . ,XN∈(L)dare affinely independent. We callN∈Nthe dimension ofS.
Remark . In a conditional simplexS=conv(X, . . . ,XN), the coefficients of convex com-binations are unique in the sense that ifNi=λi=Ni=μi= , then
N
i= λiXi=
N
i=
Indeed, sinceNi=(λi–μi)Xi= and
N
i=(λi–μi) = , it follows from (.) thatλi–μi= 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 ofLpor the convergence in theLp-norm for ≤p<∞. First,Lp 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, . . . ,XN) in (L)dsuch that anyXk is in (Lp)d, it holds thatSis uniformly bounded byNsupk=,...,NXk ∈Lp. This uniform boundedness yields that anyP-almost sure converging sequence inS is also converging in theLp-norm for ≤p<∞due to the dominated convergence theorem. This shows how one can translate results fromLtoLp.
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 . Let(Xi)Ni=and(Yi)Ni=be families in(L)dwithσ(X, . . . ,XN) =σ(Y, . . . ,
YN).Thenconv(X, . . . ,XN) =conv(Y, . . . ,YN).Moreover, (Xi)Ni=are affinely independent if
and only if(Yi)Ni=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 (Xi)Ni= be affinely independent andσ(X, . . . ,XN) =σ(Y, . . . ,YN). We want to show that (Yi)Ni=are affinely independent. To that end, we define the affine hull
aff(X, . . . ,XN) =
N
i=
λiXi:λi∈L, N
i= λi=
.
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. SinceXi∈σ(X, . . . ,XN) =σ(Z, . . . ,ZM)⊆aff(Z, . . . ,ZM), we have
aff(X, . . . ,XN)⊆aff(Z, . . . ,ZM). Further, it holds thatX=
M
i=BiZifor a partition (Bi)Mi=
and hence there exists at least oneBksuch thatAk:=Bk∩A∈A+, and Ak
X= Ak
Zk. Therefore,
A
kaff(X, . . . ,XN)⊆Akaff(Z, . . . ,ZM)
= A
kaff {X,Z, . . . ,ZM} \ {Zk}
ForX=
M
i=AiZi, we find a setAksuch thatAk =Ak∩Ak∈A+, Ak
X= Ak
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)Ni=. Hence, we can again
substituteZk byXonAk. Inductively, we findk, . . . ,kN such that
AkNaff(X, . . . ,XN)⊆AkNaff {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)Ni=such that
N i=λiYi=
N
i=λi= but not all coefficientsλi are zero, without loss of generality,λ> onA∈A+. Thus, AY= –ANi=
λi
λYiand it holds that Aaff(Y, . . . ,YN) = Aaff(Y, . . . ,YN). To see this, consider AZ= A
N
i=μiYiin Aaff(Y, . . . ,YN), which means A
N
i=μi= A. Thus, inserting forY,
AZ= A
N
i=
μiYi–μ
N i= λi λ Yi
= A
N
i=
μi–μ λi λ Yi . Moreover, A N i=
μi–μ λi
λ
= A
N
i= μi
+ A
–μ
λ N i= λi
= A( –μ) + A
μ λ
λ= A.
Hence, it holds that AZ∈Aaff(Y, . . . ,YN). It follows that Aaff(X, . . . ,XN) = 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, . . . ,XN) 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, . . . ,XN) which isX=
N
k=AkXkfor a partition
(Ak)k=,...,N. Now assume to the contrary that we findY=
N
k=λkXkandZ=
N
k=μkXkin
S\ {X}such thatX=λY+ ( –λ)Z. This means thatX=Nk=(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 thatλλk+ ( –λ)μk= Akif and only if
λk=μk= 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=νkXksuch that there existνkandνkandB∈A+with <νk< onBand <νk< onB. Defineε:=
ess inf{νk,νk, –νk, –νk}. Then defineμk=λk=νkifk=k=kandλk=νk–ε,λk=
νk+ε,μk=νk+εandμk=νk–ε. Thus,Y=
N
k=λkXkandZ=
N
k=μkXkfulfill .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, . . . ,XN)⊆
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
As an example, let us consider [, ](A). For an arbitraryA∈A, it holds that Aand Acare affinely independent andconv(A, Ac) ={λA+ ( –λ)Ac: ≤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, Ac) for allA∈A.
Remark . In (L)d, lete
ibe 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, . . . ,XN) be a conditional simplex. We define the set of
extremal pointsext(S) =σ(X, . . . ,XN). For an index setIand a collectionS = (Si)i∈I of conditional simplexes, we denoteext(S) =σ(i∈Iext(Si)).
Remark . LetSj=conv(Xj
, . . . ,X
j
N),j∈N, be conditional simplexes of the same dimen-sionNand (Aj)j∈Na partition. Thenj∈NAjSjis again a conditional simplex. To that end,
we defineYk=
j∈NAjX
j
kand recognize
j∈NAjS
j=conv(Y
, . . . ,YN). Indeed,
N
k= λkYk=
N
k= λk
j∈N
AjX
j k=
j∈N
Aj
N
k= λkXjk∈
j∈N
AjS
j, (.)
showsconv(Y, . . . ,YN)⊆
j∈NAjS
j. The other inclusion follows by consideringN k=λ
j k×
Xkj∈Sjand definingλ k=
j∈NAjλ
j
k. To show thatY, . . . ,YNare affinely independent, we considerNk=λkYk= =
N
k=λk. Then by (.) it holds that Aj
N
k=λkXkj= and since
Sjis a conditional simplex,
Ajλk= for allj∈Nandk= , . . . ,N. From the fact that (Aj)j∈N
is a partition, it follows thatλk= 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
Ykπ= k
k
i=
Xπ(i), k= , . . . ,N, Cπ=conv Yπ, . . . ,YNπ
.
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
(i) σ(π∈SNCπ) =S.
(ii) Cπ has dimensionN,π∈SN.
(iii) Cπ∩Cπis a conditional simplex of dimensionr∈Nandr<Nforπ,π∈SN,π=π. (iv) Fors= , . . . ,N– ,letBs:=conv(X, . . . ,Xs).All conditional simplexesCπ∩Bs,
π∈SN,of dimensionssubdivideBsbarycentrically.
Proof We show the affine independence ofYπ
, . . . ,YNπ inCπ. It holds that
λπ()Xπ()+λπ()
Xπ()+Xπ()
+· · ·+λπ(N)
N k=Xπ(k)
N =
N
i= μiXi,
withμi=
N k=π–(i)
λπ(k)
k . Since
N i=μi=
N
i=λi, the affine independence ofYπ, . . . ,YNπ is obtained by the affine independence ofX, . . . ,XN. Therefore allCπ are conditional
sim-plexes.
As for Condition (i), it clearly holds thatσ(π∈S
NCπ)⊆S. On the other hand, letX=
N
i=λiXi∈S. Then we find a partition (An)n=,...,M, for someM∈N, such that on every
Anthe indexes are completely ordered, which isλin
≥λin ≥ · · · ≥λinN onAn.
aThis means
thatX∈AnCπnwithπn(j) =inj. Indeed, we can rewriteXonAnas
X= (λin
–λin)Xin+· · ·+ (N– )(λinN––λinN)
N–
k= Xink
N– +NλinN
N k=Xink
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 Yjπ:j∈J
. (.)
To show ⊇, let j∈ J. It holds that Yπ
j 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=Ni=λi(
i k=
Xπ(k) i ) =
N i=μi(
i k=
Xπ(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
λi
i =
N i=π–(π(p))
μi
i. The same holds forXπ(q):
N i=q
μi
i =
N
i=π–(π(q))λii. Put together N
i=j+ μi
i ≤
N
i=q
μi
i =
N
i=π–(π(q)) λi
i ≤
N
i=j+ λi
i ≤
N
i=p
λi
i =
N
i=π–(π(p)) μi
i ≤
N
i=j+ μi
i ,
which is only possible ifμj=λj= 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=Ni=λiXiwithλi> for alli= , . . . ,sandλi= fori>s. As an element ofCπ, thisYhas a representation of the
formY=Nj=(Nk=jμk
k )Xπ(j)for
N
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 thatNk=jμk
k = and within
N k=j
μk
k = for allj≥j. This meansY=
j–
j= (
N k=j
μk
k)Xπ(j)and henceY is the convex combination ofj– elements withj– <s. This contradicts the property thatλi> 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 bySm. Further, we defineext(Sm) =σ({ext(C) :C∈Sm}) 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(Yπ, . . . ,YNπ),π∈SNin the barycentric sub-division of a conditional simplexS. Then it holds that
diam(Cπ) =ess sup
i,j=,...,N
Yiπ–Yjπ≤N–
N diam(S).
Since this holds for any π ∈SN, it follows that the diameter ofSm, which is an arbi-trary conditional simplex of them-fold barycentric subdivision ofS, fulfillsdiam(Sm)≤ (NN–)mdiam(S). Since diam(S) < ∞ and (N–
N )m → , for m → ∞, it follows that
diam(Sm)→ form→ ∞for every sequence (Sm) m∈N.
2 Brouwer fixed point theorem for conditional simplexes
Definition . LetS=conv(X, . . . ,XN) be a conditional simplex,m-fold barycentrically subdivided inSm. A local functionφ:ext(Sm)→ {, . . . ,N}(A) is called alabeling func-tionofS. For fixedX, . . . ,XN ∈ext(S) withS=conv(X, . . . ,XN), the labeling function is calledproperif for anyY∈ext(Sm) it holds that
P ω:φ(Y)(ω) =i,λi(ω) =
=
fori= , . . . ,N, whereY=Ni=λiXi. A conditional simplexC=conv(Y, . . . ,YN), withC⊆
S, withYj∈ext(Sm),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, . . . ,XN)be a conditional simplex and f :S →S be a
(i) P({ω:φ(X)(ω) =i;λi(ω) = orμi(ω) >λi(ω)}) = for alli= , . . . ,N, (ii) P({ω:φ(X)(ω) = ,∃i∈ {, . . . ,N},λi(ω) > ,λi(ω)≥μi(ω)}) = ,
where(λi)i=,...,Nand(μi)i=,...,Nare determined by X=
N
i=λiXiand f(X) =
N
i=μiXi.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},λi(ω)≥μi(ω),λi(ω) >
= .
Assume, to the contrary, thatμi>λionA∈A+for allλiwithλi> onA. Then it holds that =Ni=λi{λi>}<
N
i=μi{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,λi(ω) = }) = , which shows thatφis proper.
To prove the existence forX∈ext(Sm) withX=N
i=λiXi,f(X) =
N
i=μi, letBi:={ω:
λi(ω) > } ∩ {ω:λi(ω)≥μi(ω)},i= , . . . ,N. Then we define the functionφ atXas{ω:
φ(X)(ω) =i}=Bi\(
i–
k=Bk),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∈NAjXj,
where Xj=Ni=λjiXi andf(Xj) =
N i=μ
j
iXi. Due to uniqueness of the coefficients in a conditional simplex, it holds thatλi=
j∈NAjλ
j
i, and due to locality off, it follows that
μi=
j∈NAjμ
j
i. Therefore it holds thatBi=
j∈N({ω:λ j
i(ω) > } ∩ {ω:λ j i(ω)≥μ
j i(ω)} ∩
Aj) =
j∈N(B j
i∩Aj). Hence,φ(X) =ionBi\(
i–
k=Bk) = [
j∈N(B j
i∩Aj)]\[
i–
k=(
j∈NB j k∩
Aj)] =
j∈N[(Bji\
i–
k=B
j
k)∩Aj]. On the other hand, we see that
j∈NAjφ(Xj) ision any Aj∩ {ω:φ(Xj)(ω) =i}, hence it isi on
j∈N(Bji\
i–
k=B
j
k)∩Aj. Thus,
j∈NAjφ(Xj) =
φ(j∈NAjXj), 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 setsλi≥μi. 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
partition (Ak)k=,...,Ksuch that on anyAkthere is an odd number of completely labeledCπ.
The caseN= is clear since a point can be labeled with the constant index only. Suppose that the caseN– is proven. Since the number ofYπ
i of the barycentric subdi-vision is finite andφcan only take finitely many values, it holds for allV∈(Yiπ)i=,...,N,π∈SN
that there exists a partition (AVk)k=,...,K,K<∞, whereφ(V) is constant on anyAVk. There-fore, we find a partition (Ak)k=,...,Ksuch thatφ(V) onAkis constant for allVandAk. Fix
Aknow.
In the following, we denote byCπbthose conditional simplexes for whichCπb∩BN–are N– -dimensional (cf.Lemma .(iv)), thereforeπb(N) =N. Further we denote byCπc
these conditional simplexes which are not of the typeCπb, that is,πc(N)=N. If we useCπ,
we mean a conditional simplex of arbitrary type. We define:
• C⊆(Cπ)π∈SN to be the set ofCπ which are completely labeled onAk.
• A ⊆(Cπ)π∈SN to be the set ofP-almost completely labeledCπ, that is,
φ Ykπ,k∈ {, . . . ,N}={, . . . ,N– } onAk.
• Eπ to be the set of intersections(Cπ∩Cπl)πl∈SN which areN– -dimensional and
completely labeled onAk.b
• Bπ to be the set of intersectionsCπ∩BN–which are completely labeled onAk. It holds thatEπ∩Bπ=∅and hence|Eπ∪Bπ|=|Eπ|+|Bπ|. SinceCπc∩BN–is at mostN–
-dimensional, it holds thatBπc=∅and hence|Bπc|= . Moreover, we know thatCπ∩Cπ l
isN– -dimensional onAkif and only if this holds on the whole(cf.Lemma .(iii)) andCπb∩BN–=∅onAkif and only if this also holds on the whole(cf.Lemma .(iv)). So, it does not play any role if we look at these sets which are intersections onAkor on since they are exactly the same sets.
IfCπc∈C, then|Eπc|= and ifCπb∈C, then|Eπb∪Bπb|= . IfCπc∈A, then|Eπc|=
and ifCπb ∈A, then|Eπb ∪Bπb|= . Therefore it holds that
π∈SN|Eπ ∪Bπ|=|C|+
|A|.
If we pickEπ∈Eπ, we know that there always exists exactly one otherπlsuch thatEπ∈
Eπl (Lemma .(iii)). Therefore
π∈SN|Eπ|is even. Moreover, (Cπb∩BN–)πbsubdivides
BN–barycentrically, and hence we can apply the inductive hypothesis (onext(Cπb∩BN–)).
Indeed, the setBN–is aσ-stable set, so if it is partitioned by the labeling function into
(Ak)k=,...,K, we know thatBN–(S) =
K
k=AkBN–(AkS) and by Lemma .(iv) we can
ap-ply the induction hypothesis also to everyAk,k= , . . . ,K. Thus, the number of completely labeled conditional simplexes is odd on a partition of, but sinceφis constant onAk, it also has to be odd there. This means thatπb|Bπb|has to be odd. Hence, we also have
thatπ|Eπ∪Bπ|is the sum of an even and an odd number and thus odd. So, we conclude
|C|+ |A|is odd and hence also|C|. Thus, we find for anyAka completely labeledCπk.
We defineS=K
k=AkCπkwhich by Remark . is indeed a conditional simplex. Due
toσ-stability ofS, it holds thatS⊆S. By Remark .,Shas a diameter which is less
thanNN–diam(S) and sinceφis local,Sis completely labeled on the whole.
The same argumentation holds for everym-fold barycentric subdivisionSmofS,m∈
N, that is, there exists a completely labeled conditional simplex in everym-fold barycen-trically subdivided conditional simplex which is properly labeled. Henceforth, subdividing
Sm+⊆S form∈N. Moreover, sinceSis completely labeled, it holdsS=K k=AkCπk
as above, whereCπk is completely labeled onAk. This meansCπk =conv(Y
k
, . . . ,YNk) with
φ(Yjk) =jonAkfor everyj= , . . . ,N. DefiningVj=
K k=AkY
k
j for everyj= , . . . ,Nyields
P({ω:φ(Vj)(ω) =j}) = for everyj= , . . . ,N andS=conv(V, . . . ,VN). The same holds for anym∈Nand so that we can writeSm=conv(Vm
, . . . ,VNm) withP({ω:φm–(Vjm)(ω) =
j}) = for everyj= , . . . ,N. Now, (Vm
)m∈Nis a sequence in the sequentially closed,L-bounded setS, so that by [,
Corollary .], there existsY∈Sand a sequence (Mm)m∈NinN(A) such thatMm+>Mm for allm∈Nandlimm→∞VMm =Y P-almost surely. ForMm∈N(A),VMm is defined as
n∈N{Mm=n}V
n
. This means an element with indexMm, for somem∈N, equalsVnon
An,n∈N, where the setsAnare determined byMmviaAn={ω:Mm(ω) =n},n∈N. Fur-thermore, asmgoes to∞,diam(Sm) is converging to zeroP-almost surely, and therefore it also follows thatlimm→∞VkMm=Y P-almost surely for everyk= , . . . ,N. Indeed, it holds that|Vm
k –Y| ≤diam(Sm) +|Vm–Y|for everyk= , . . . ,Nandm∈N, so we can use the
sequence (Mm)m∈Nfor everyk= , . . . ,N. Let Y =Nl=αlXl and f(Y) =
N
l=βlXl as well as Vkm =
N
l=λml ,kXl and f(Vkm) =
N
l=μml,kXl form∈N. Asf is local, it holds thatf(V Mm
) =
n∈N{Mm=n}f(Vn). By
se-quential continuity of f, it follows thatlimm→∞f(VkMn) =f(Y)P-almost surely for ev-eryk= , . . . ,N. In particular,limm→∞λMl m,l=αlandlimm→∞μMl m,l=βlP-almost surely for everyl= , . . . ,N. However, by construction,φm–(Vm
l ) =lfor everyl= , . . . ,N, and from the choice ofφm–, it follows thatλml,l≥μml,l P-almost surely for everyl= , . . . ,N and m∈N. Hence, αl=limm→∞λlMm,l≥limm→∞μlMm,l =βl P-almost surely for every
l= , . . . ,N. This is possible only ifαl=βlP-almost surely for everyl= , . . . ,N, showing
thatf(Y) =Y.
3 Applications
3.1 Fixed point theorem for sequentially closed and bounded sets in (L0)d
Proposition . LetKbe an L-convex,sequentially closed and bounded subset of(L)d,
and let f :K→Kbe a local,sequentially continuous function.Then f has a fixed point.
Proof SinceKis bounded, there exists a conditional simplexS such thatK⊆S. Now
define the functionh:S→Kby
h(X) =
⎧ ⎨ ⎩
X, ifX∈K,
arg min{X–Y:Y∈K}, else.
This means, thathis the identity function onKand the projection onKfor the elements inS\K. Due to [, Corollary .] this minimum exists and is unique. Thereforehis well defined.
We can characterizehby
Y=h(X) ⇔ X–Y,Z–Y ≤ for allZ∈K. (.)
Indeed, let X–Y,Z–Y ≤ for allZ∈K. Then
X–Z=(X–Y) + (Y–Z)
which shows the minimizing property ofh. On the other hand, letY=h(X). SinceKis L-convex,λZ+ ( –λ)Y∈Kfor anyλ∈(, ](A) andZ∈K. By a standard calculation,
X– λZ+ ( –λ)Y≥ X–Y
yields ≥–λX, –Y+ (λ–λ)Y,Y+ λ X,Z–λZ– λ( –λ)Z,Y. Dividing by λ> and lettingλ↓ afterwards yields
≥– X, –Y+ Y,Y+ X,Z– Z,Y= X–Y,Z–Y,
which is the desired claim.
Furthermore, for anyX,Y∈S, it holds that
h(X) –h(Y)≤ X–Y.
Indeed,
X–Y= h(X) –h(Y)+X–h(X) +h(Y) –Y=: h(X) –h(Y)+c,
which means
X–Y=h(X) –h(Y)+c+ c,h(X) –h(Y). (.)
Since
c,h(X) –h(Y)= –X–h(X),h(Y) –h(X)–Y–h(Y),h(X) –h(Y),
by (.), it follows that c,h(X) –h(Y) ≥. Therefore,X–Y≥ h(X) –h(Y)by (.).
This shows thathis sequentially continuous.
The function f ◦his a sequentially continuous function mapping fromS to K⊆S. Hence, there exists a fixed pointf ◦h(Z) =Z. Sincef ◦hmaps intoK, thisZhas to be inK. But then we knowh(Z) =Zand thereforef(Z) =Z, which ends the proof.
Remark . In Drapeauet al.[] the concept of conditional compactness is introduced and it is shown that there is an equivalence between conditional compactness and condi-tional closed- and boundedness in (L)d. In that context we can formulate the conditional Brouwer fixed point theorem as follows. A sequentially continuous functionf :K→K such thatKis a conditionally compact andL-convex subset of (L)dhas a fixed point.
3.2 Applications in conditional analysis on (L0)d
Working inRd, the Brouwer fixed point theorem can be used to prove several topological properties and is even equivalent to some of them. In the theory of (L)d, we will show that the conditional Brouwer fixed point theorem has several implications as well.
Definition . LetX andYbe subsets of (L)d. AnL-homotopyof two local,
sequen-tially continuous functionsf,g:X→Yis a jointly local, sequentially continuous function H:X ×[, ](A)→Y such thatH(X, ) =f(X) andH(X, ) =g(X). Jointly local means H(j∈NAjXj,
j∈NAjtj) =
j∈NAjH(Xj,tj) for any partition (Aj)j∈N, (Xj)j∈N in X and
(tj)j∈N in [, ](A). Sequential continuity ofHis thereforeH(Xn,tn)→H(X,t) whenever
Xn→Xandtn→tbothP-almost surely forXn,X∈Xandtn,t∈[, ](A).
Lemma . The identity function of the sphere is not L-homotopic to a constant function.
The proof is a consequence of the following lemma.
Lemma . There does not exist a local,sequentially continuous function f :B(d)→S(d– )which is the identity onS(d– ).
Proof Suppose that there is this local, sequentially continuous functionf. Define the func-tiong:S(d– )→S(d– ) by g(X) = –X. Then the compositiong◦f :B(d)→B(d), which actually maps toS(d– ), is local and sequentially continuous. Therefore, this has a fixed pointY which has to be inS(d– ) since this is the image ofg◦f. But we know f(Y) =Y andg(Y) = –Y and henceg◦f(Y) = –Y. Therefore,Y cannot be a fixed point
(since /∈S(d– )), which is a contradiction.
It directly follows that the identity on the sphere is notL-homotopic to a constant
func-tion. In the cased= , we get the following result which is theL-module version of an
intermediate value theorem.
Lemma . Let X,X∈Lwith X≤X and[X,X] :={Z∈L:X≤Z≤X}.Let further f : [X,X]→Lbe a local,sequentially continuous function and A:={ω:f(X)(ω)≤f(X)(ω)}.
If Y is in[Af(X) + Acf(X), Af(X) + Acf(X)],then there exists Y∈[X,X]with f(Y) =Y.
Proof Sincef is local, it is sufficient to prove the case forf(X)≤f(X) which isA=. For the general case, we would considerAandAcseparately, obtain
Af(Y) = AY, Acf(Y) =
AcYand by locality we havef(AY+ AcY) =Y. So, suppose thatYis in [f(X),f(X)] in
the rest of the proof.
Let firstf(X) <Y<f(X). Define the functiong: [X,X]→[X,X] by
g(V) :=pV–f(V) +Y with p(Z) = {Z≤X}X+ {X≤Z≤X}Z+ {X≤Z}X.
Notice that as a sum, product, and composition of local, sequentially continuous func-tions,gis so as well. Hence,ghas a fixed pointY. IfY=X, it must hold thatX–f(X) +Y≤ X, which meansY ≤f(X), which is a contradiction. IfY =X, it follows thatf(X)≤Y, which is also a contradiction. Hence,Y=Y–f(Y) +Y, which meansf(Y) =Y.
IfY=f(X) onBandY=f(X) onC, thenf(X) <Y<f(X) onD:= (B∪C)c. Then we find
Ysuch thatf(Y) =YonD. In totalf(BX+ C\BX+ DY) = Bf(X) + C\Bf(X) + Df(Y) =Y. This shows the claim for generalYin [f(X),f(X)].
Competing interests
Authors’ contributions
All authors have contributed in equal parts to the paper. All authors read and approved the final manuscript.
Author details
1Humboldt-Universität zu Berlin, Unter den Linden 6, Berlin, 10099, Germany.2Universität Konstanz, Universitätsstraße 10, Konstanz, 78464, Germany.
Acknowledgements
We thank Asgar Jamneshan for fruitful discussions. The first author was supported by MATHEON project E11. The second author was supported by Konsul Karl und Dr. Gabriele Sandmann Stiftung. The fourth author was supported by Evangelisches Studienwerk Villigst.
Endnotes
a LetB
π:={ω:λπ(1)(ω)≥λπ(2)(ω)≥ · · · ≥λπ(N)(ω)},π∈SN. This finite collection of measurable sets fulfills
P(π∈SNBπ) = 1. We can construct a partition (An)n=1,...,Msuch thatAn⊆Bπnfor someπn∈SNand for alln= 1,. . .,M.
Such a partition fulfills the required property. b That is bearing exactly the label 1,. . .,N– 1 onA
k.
Received: 17 May 2013 Accepted: 21 October 2013 Published:19 Nov 2013
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