Open Access

Brouwer fixed point theorem in ( L 0 ) d

  • Samuel Drapeau1,
  • Martin Karliczek1,
  • Michael Kupper2Email author and
  • Martin Streckfuß1
Fixed Point Theory and Applications20132013:301

https://doi.org/10.1186/1687-1812-2013-301

Received: 17 May 2013

Accepted: 21 October 2013

Published: 19 November 2013

Abstract

The classical Brouwer fixed point theorem states that in R d every continuous function from a convex, compact set on itself has a fixed point. For an arbitrary probability space, let L 0 = L 0 ( Ω , A , P ) be the set of random variables. We consider ( L 0 ) d as an L 0 -module and show that local, sequentially continuous functions on L 0 -convex, closed and bounded subsets have a fixed point which is measurable by construction.

MSC:47H10, 13C13, 46A19, 60H25.

Keywords

conditional simplexfixed points in ( L 0 ) d

Introduction

The Brouwer fixed point theorem states that a continuous function from a compact and convex set in R d 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 et al. [1], inspired by the theory developed by Filipović et al. [2] and Guo [3], studied ( L 0 ) d as an L 0 -module, discussing concepts like linear independence, σ-stability, locality and L 0 -convexity. Based on this, we define affine independence and conditional simplexes in ( L 0 ) 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 suitable σ-combination of elements of the barycentric subdivision along a partition of Ω. 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 construction. Hence, even though we follow the constructions and methods used in the proof of the classical result in R d (cf. [4]), 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. Given C , a compact convex set of a Banach space, a continuous random operator is a function R : Ω × C C satisfying
  1. (i)

    R ( , x ) : Ω C is a random variable for any fixed x C ,

     
  2. (ii)

    R ( ω , ) : C C is a continuous function for any fixed ω Ω .

     

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 any ω (cf. [57]). In contrast to this ω-wise consideration, our approach is completely within the theory of L 0 . All objects and properties are therefore defined in that language and proofs are done with L 0 -methods. Moreover, the connection between continuous random operators on R d and sequentially continuous functions on ( L 0 ) d is not entirely clear.

An application, though not studied in this paper, is for instance possible in economic theory or optimization in the context of [8]. Therein the methods from convex analysis are used to obtain equilibrium results for translation invariant utility functionals on ( L 0 ) 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 to L 0 .

The present paper is organized as follows. In the first section, we present the basic concepts concerning ( L 0 ) d as an L 0 -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 for L 0 -convex, bounded and sequentially closed sets in ( L 0 ) 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 0 ) d and, on the other hand, an intermediate value theorem in L 0 .

1 Conditional simplex

For a probability space ( Ω , A , P ) , let L 0 = L 0 ( Ω , A , P ) be the space of all A -measurable random variables, where P-almost surely equal random variables are identified. In particular, for X , Y L 0 , the relations X Y and X > Y have to be understood P-almost surely. The set L 0 with the P-almost everywhere order is a lattice ordered ring, and for a non-empty subset C L 0 , we denote the least upper bound by ess sup C and the greatest lower bound by ess inf C , respectively (cf. [1]). For m R , we denote the constant random variable m 1 Ω by m. Further, we define the sets L + 0 = { X L 0 : X 0 } , L + + 0 = { X L 0 : X > 0 } and A + = { A A : P ( A ) > 0 } . The set of random variables with values in a set M R is denoted by M ( A ) . For example, { 1 , , r } ( A ) is the set of A -measurable functions with values in { 1 , , r } N , [ 0 , 1 ] ( A ) = { Z L 0 : 0 Z 1 } and ( 0 , 1 ) ( A ) = { Z L 0 : 0 < Z < 1 } .

The convex hull of X 1 , , X N ( L 0 ) d , N N , is defined as
conv ( X 1 , , X N ) = { i = 1 N λ i X i : λ i L + 0 , i = 1 N λ i = 1 } .

An element Y = i = 1 N λ i X i such that λ i > 0 for all i I { 1 , , N } is called a strict convex combination of { X i : i I } . Moreover, a set C ( L 0 ) d is said to be L 0 -convex if for any X , Y C and λ [ 0 , 1 ] ( A ) , it holds that λ X + ( 1 λ ) Y C .

The σ-stable hull of a set C ( L 0 ) d is defined as
σ ( C ) = { i N 1 A i X i : X i C , ( A i ) i N  is a partition } ,
where a partition is a countable family ( A i ) i N A such that P ( A i A j ) = 0 for i j and P ( i N A i ) = 1 . We call a non-empty set C σ-stable if it is equal to σ ( C ) . For a σ-stable set C ( L 0 ) d , a function f : C ( L 0 ) d is called local if f ( i N 1 A i X i ) = i N 1 A i f ( X i ) for every partition ( A i ) i N and X i C , i N . For X , Y ( L 0 ) d , we call a function f : X Y sequentially continuous if for every sequence ( X n ) n N in X converging to X X P-almost-surely, it holds that f ( X n ) converges to f ( X ) P-almost surely. Further, the L 0 -scalar product and L 0 -norm on ( L 0 ) d are defined as
X , Y = i = 1 d X i Y i and X = X , X 1 2 .

We call C ( L 0 ) d bounded if ess sup X C X L 0 and sequentially closed if it contains all P-almost sure limits of sequences in C . Further, the diameter of C ( L 0 ) d is defined as diam ( C ) = ess sup X , Y C X Y .

Definition 1.1 Elements X 1 , , X N of ( L 0 ) d , N N , are said to be affinely independent, if either N = 1 or N > 1 and { X i X N } i = 1 N 1 are linearly independent, that is,
i = 1 N 1 λ i ( X i X N ) = 0 implies λ 1 = = λ N 1 = 0 ,
(1.1)

where λ 1 , , λ N 1 L 0 .

The definition of affine independence is equivalent to
i = 1 N λ i X i = 0 and i = 1 N λ i = 0 implies λ 1 = = λ N = 0 .
(1.2)

Indeed, first we show that (1.1) implies (1.2). Let i = 1 N λ i X i = 0 and i = 1 N λ i = 0 . Then i = 1 N 1 λ i ( X i X N ) = λ N X N + i = 1 N 1 λ i X i = 0 . By assumption (1.1), it holds that λ 1 = = λ N 1 = 0 , thus also λ N = 0 . To see that (1.2) implies (1.1), let i = 1 N 1 λ i ( X i X N ) = 0 . With λ N = i = 1 N 1 λ i , it holds that i = 1 N λ i X i = λ N X N + i = 1 N 1 λ i X i = i = 1 N 1 λ i ( X i X N ) = 0 . By assumption (1.2), λ 1 = = λ N = 0 .

Remark 1.2 We observe that if ( X i ) i = 1 N ( L 0 ) d are affinely independent, then ( λ X i ) i = 1 N for λ L + + 0 and ( X i + Y ) i = 1 N for Y ( L 0 ) d are affinely independent. Moreover, if a family X 1 , , X N is affinely independent, then also 1 B X 1 , , 1 B X N are affinely independent on B A + , which means from i = 1 N 1 B λ i X i = 0 and i = 1 N 1 B λ i = 0 it always follows that 1 B λ i = 0 for all i = 1 , , N .

Definition 1.3 A conditional simplex in ( L 0 ) d is a set of the form
S = conv ( X 1 , , X N )

such that X 1 , , X N ( L 0 ) d are affinely independent. We call N N the dimension of S .

Remark 1.4 In a conditional simplex S = conv ( X 1 , , X N ) , the coefficients of convex combinations are unique in the sense that if i = 1 N λ i = i = 1 N μ i = 1 , then
i = 1 N λ i X i = i = 1 N μ i X i implies λ i = μ i for all  i = 1 , , N .
(1.3)

Indeed, since i = 1 N ( λ i μ i ) X i = 0 and i = 1 N ( λ i μ i ) = 0 , it follows from (1.2) that λ i μ i = 0 for all i = 1 , , N .

Remark 1.5 Note that the present setting - L 0 -modules and the sequential P-almost sure convergence - is of local nature. This is, for instance, not the case for subsets of L p or the convergence in the L p -norm for 1 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 a σ-stable subspace of L 0 . However, for a conditional simplex S = conv ( X 1 , , X N ) in ( L 0 ) d such that any X k is in ( L p ) d , it holds that S is uniformly bounded by N sup k = 1 , , N X k L p . This uniform boundedness yields that any P-almost sure converging sequence in S is also converging in the L p -norm for 1 p < due to the dominated convergence theorem. This shows how one can translate results from L 0 to L p .

Since a conditional simplex is a convex hull, it is in particular σ-stable. In contrast to a simplex in R d , the representation of S as a convex hull of affinely independent elements is unique but up to σ-stability.

Proposition 1.6 Let ( X i ) i = 1 N and ( Y i ) i = 1 N be families in ( L 0 ) d with σ ( X 1 , , X N ) = σ ( Y 1 , , Y N ) . Then conv ( X 1 , , X N ) = conv ( Y 1 , , Y N ) . Moreover, ( X i ) i = 1 N are affinely independent if and only if ( Y i ) i = 1 N are affinely independent.

If S is a conditional simplex such that S = conv ( X 1 , , X N ) = conv ( Y 1 , , Y N ) , then it holds that σ ( X 1 , , X N ) = σ ( Y 1 , , Y N ) .

Proof Suppose σ ( X 1 , , X N ) = σ ( Y 1 , , Y N ) . For i = 1 , , N , it holds that
X i σ ( X 1 , , X N ) = σ ( Y 1 , , Y N ) conv ( Y 1 , , Y N ) .

Therefore, conv ( X 1 , , X N ) conv ( Y 1 , , Y N ) and the reverse inclusion holds analogously.

Now, let ( X i ) i = 1 N be affinely independent and σ ( X 1 , , X N ) = σ ( Y 1 , , Y N ) . We want to show that ( Y i ) i = 1 N are affinely independent. To that end, we define the affine hull
aff ( X 1 , , X N ) = { i = 1 N λ i X i : λ i L 0 , i = 1 N λ i = 1 } .
First, let Z 1 , , Z M ( L 0 ) d , M N , such that σ ( X 1 , , X N ) = σ ( Z 1 , , Z M ) . We show that if 1 A aff ( X 1 , , X N ) 1 A aff ( Z 1 , , Z M ) for A A + and X 1 , , X N are affinely independent, then M N . Since X i σ ( X 1 , , X N ) = σ ( Z 1 , , Z M ) aff ( Z 1 , , Z M ) , we have aff ( X 1 , , X N ) aff ( Z 1 , , Z M ) . Further, it holds that X 1 = i = 1 M 1 B i 1 Z i for a partition ( B i 1 ) i = 1 M and hence there exists at least one B k 1 1 such that A k 1 1 : = B k 1 1 A A + , and 1 A k 1 1 X 1 = 1 A k 1 1 Z k 1 . Therefore,
1 A k 1 1 aff ( X 1 , , X N ) 1 A k 1 1 aff ( Z 1 , , Z M ) = 1 A k 1 1 aff ( { X 1 , Z 1 , , Z M } { Z k 1 } ) .
For X 2 = i = 1 M 1 A i 2 Z i , we find a set A k 2 such that A k 2 2 = A k 2 A k 1 1 A + , 1 A k 2 2 X 2 = 1 A k 2 2 Z k 2 and k 1 k 2 . Assume to the contrary k 2 = k 1 , then there exists a set B A + such that 1 B X 1 = 1 B X 2 , which is a contradiction to the affine independence of ( X i ) i = 1 N . Hence, we can again substitute Z k 2 by X 2 on A k 2 2 . Inductively, we find k 1 , , k N such that
1 A k N aff ( X 1 , , X N ) 1 A k N aff ( { X 1 , , X N , Z 1 , , Z M } { Z k 1 , , Z k N } ) ,
which shows M N . Now suppose that Y 1 , , Y N are not affinely independent. This means that there exist ( λ i ) i = 1 N such that i = 1 N λ i Y i = i = 1 N λ i = 0 but not all coefficients λ i are zero, without loss of generality, λ 1 > 0 on A A + . Thus, 1 A Y 1 = 1 A i = 2 N λ i λ 1 Y i and it holds that 1 A aff ( Y 1 , , Y N ) = 1 A aff ( Y 2 , , Y N ) . To see this, consider 1 A Z = 1 A i = 1 N μ i Y i in 1 A aff ( Y 1 , , Y N ) , which means 1 A i = 1 N μ i = 1 A . Thus, inserting for Y 1 ,
1 A Z = 1 A [ i = 2 N μ i Y i μ 1 i = 2 N λ i λ 1 Y i ] = 1 A [ i = 2 N ( μ i μ 1 λ i λ 1 ) Y i ] .
Moreover,
1 A [ i = 2 N ( μ i μ 1 λ i λ 1 ) ] = 1 A [ i = 2 N μ i ] + 1 A [ μ 1 λ 1 i = 2 N λ i ] = 1 A ( 1 μ 1 ) + 1 A μ 1 λ 1 λ 1 = 1 A .

Hence, it holds that 1 A Z 1 A aff ( Y 2 , , Y N ) . It follows that 1 A aff ( X 1 , , X N ) = 1 A aff ( Y 1 , , Y N ) = 1 A aff ( Y 2 , , Y N ) . This is a contradiction to the former part of the proof (because N 1 N ).

Next, we show that in a conditional simplex S = conv ( X 1 , , X N ) it holds that X is in σ ( X 1 , , X N ) if and only if there do not exist Y and Z in S { X } and λ ( 0 , 1 ) ( A ) such that λ Y + ( 1 λ ) Z = X . Consider X σ ( X 1 , , X N ) which is X = k = 1 N 1 A k X k for a partition ( A k ) k = 1 , , N . Now assume to the contrary that we find Y = k = 1 N λ k X k and Z = k = 1 N μ k X k in S { X } such that X = λ Y + ( 1 λ ) Z . This means that X = k = 1 N ( λ λ k + ( 1 λ ) μ k ) X k . Due to uniqueness of the coefficients (cf. (1.3)) in a conditional simplex, we have λ λ k + ( 1 λ ) μ k = 1 A k for all k = 1 , , N . By means of 0 < λ < 1 , it holds that λ λ k + ( 1 λ ) μ k = 1 A k if and only if λ k = μ k = 1 A k . Since the last equality holds for all k, it follows that Y = Z = X . Therefore, we cannot find Y and Z in S { X } such that X is a strict convex combination of them. On the other hand, consider X S such that X σ ( X 1 , , X N ) . This means X = k = 1 N ν k X k such that there exist ν k 1 and ν k 2 and B A + with 0 < ν k 1 < 1 on B and 0 < ν k 2 < 1 on B. Define ε : = ess inf { ν k 1 , ν k 2 , 1 ν k 1 , 1 ν k 2 } . Then define μ k = λ k = ν k if k 1 k k 2 and λ k 1 = ν k 1 ε , λ k 2 = ν k 2 + ε , μ k 1 = ν k 1 + ε and μ k 2 = ν k 2 ε . Thus, Y = k = 1 N λ k X k and Z = k = 1 N μ k X k fulfill 0.5 Y + 0.5 Z = X but both are not equal to X by construction. Hence, X can be written as a strict convex combination of elements in S { X } . To conclude, consider X σ ( X 1 , , X N ) S = conv ( X 1 , , X N ) = conv ( Y 1 , , Y N ) . Since X σ ( X 1 , , X N ) , it is not a strict convex combination of elements in S { X } , in particular, of elements in conv ( Y 1 , , Y N ) { X } . Therefore, X is also in σ ( Y 1 , , Y N ) . Hence, σ ( X 1 , , X N ) σ ( Y 1 , , Y N ) . With the same argumentation, the other inclusion follows. □

As an example, let us consider [ 0 , 1 ] ( A ) . For an arbitrary A A , it holds that 1 A and 1 A c are affinely independent and conv ( 1 A , 1 A c ) = { λ 1 A + ( 1 λ ) 1 A c : 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 L 0 . This is due to the fact that σ ( 0 , 1 ) = { 1 B : B A } = σ ( 1 A , 1 A c ) for all A A .

Remark 1.7 In ( L 0 ) d , let e i be the random variable which is 1 in the i th component and 0 in any other. Then the family 0 , e 1 , , e d is affinely independent and ( L 0 ) d = aff ( 0 , e 1 , , e d ) . Hence, the maximal number of affinely independent elements in ( L 0 ) d is d + 1 .

The characterization of X σ ( X 1 , , X N ) leads to the following definition.

Definition 1.8 Let S = conv ( X 1 , , X N ) be a conditional simplex. We define the set of extremal points ext ( S ) = σ ( X 1 , , X N ) . For an index set I and a collection S = ( S i ) i I of conditional simplexes, we denote ext ( S ) = σ ( i I ext ( S i ) ) .

Remark 1.9 Let S j = conv ( X 1 j , , X N j ) , j N , be conditional simplexes of the same dimension N and ( A j ) j N a partition. Then j N 1 A j S j is again a conditional simplex. To that end, we define Y k = j N 1 A j X k j and recognize j N 1 A j S j = conv ( Y 1 , , Y N ) . Indeed,
k = 1 N λ k Y k = k = 1 N λ k j N 1 A j X k j = j N 1 A j k = 1 N λ k X k j j N 1 A j S j ,
(1.4)

shows conv ( Y 1 , , Y N ) j N 1 A j S j . The other inclusion follows by considering k = 1 N λ k j X k j S j and defining λ k = j N 1 A j λ k j . To show that Y 1 , , Y N are affinely independent, we consider k = 1 N λ k Y k = 0 = k = 1 N λ k . Then by (1.4) it holds that 1 A j k = 1 N λ k X k j = 0 and since S j is a conditional simplex, 1 A j λ k = 0 for all j N and k = 1 , , N . From the fact that ( A j ) j N is a partition, it follows that λ k = 0 for all k = 1 , , N .

We will prove the Brouwer fixed point theorem in the present setting using an L 0 -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 1.10 Let S = conv ( X 1 , , X N ) be a conditional simplex and S N be the group of permutations of { 1 , , N } . Then, for π S N , we define
Y k π = 1 k i = 1 k X π ( i ) , k = 1 , , N , C π = conv ( Y 1 π , , Y N π ) .

We call ( C π ) π S N the barycentric subdivision of S .

Lemma 1.11 Let X 1 , , X N ( L 0 ) d be affinely independent. The barycentric subdivision of S = conv ( X 1 , , X N ) is a collection of finitely many conditional simplexes satisfying the following properties:
  1. (i)

    σ ( π S N C π ) = S .

     
  2. (ii)

    C π has dimension N, π S N .

     
  3. (iii)

    C π C π ¯ is a conditional simplex of dimension r N and r < N for π , π ¯ S N , π π ¯ .

     
  4. (iv)

    For s = 1 , , N 1 , let B s : = conv ( X 1 , , X s ) . All conditional simplexes C π B s , π S N , of dimension s subdivide B s barycentrically.

     
Proof We show the affine independence of Y 1 π , , Y N π in C π . It holds that
λ π ( 1 ) X π ( 1 ) + λ π ( 2 ) X π ( 1 ) + X π ( 2 ) 2 + + λ π ( N ) k = 1 N X π ( k ) N = i = 1 N μ i X i ,

with μ i = k = π 1 ( i ) N λ π ( k ) k . Since i = 1 N μ i = i = 1 N λ i , the affine independence of Y 1 π , , Y N π is obtained by the affine independence of X 1 , , X N . Therefore all C π are conditional simplexes.

As for Condition (i), it clearly holds that σ ( π S N C π ) S . On the other hand, let X = i = 1 N λ i X i S . Then we find a partition ( A n ) n = 1 , , M , for some M N , such that on every A n the indexes are completely ordered, which is λ i 1 n λ i 2 n λ i N n on A n .a This means that X 1 A n C π n with π n ( j ) = i j n . Indeed, we can rewrite X on A n as
X = ( λ i 1 n λ i 2 n ) X i 1 n + + ( N 1 ) ( λ i N 1 n λ i N n ) k = 1 N 1 X i k n N 1 + N λ i N n k = 1 N X i k n N ,

which shows that X C π n on A n . Condition (ii) is fulfilled by construction.

The intersection of two conditional simplexes C π and C π ¯ can 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 both π and π ¯ have the same set of images. Then
C π C π ¯ = conv ( Y j π : j J ) .
(1.5)
To show , let j J . It holds that Y j π is in both C π and C π ¯ since { π ( 1 ) , , π ( j ) } = { π ¯ ( 1 ) , , π ¯ ( j ) } . Since the intersection of L 0 -convex sets is L 0 -convex, we get this inclusion. As for the reverse inclusion, consider X C π C π ¯ . From X C π C π ¯ , it follows that X = i = 1 N λ i ( k = 1 i X π ( k ) i ) = i = 1 N μ i ( k = 1 i X π ¯ ( k ) i ) . Consider j J . By definition of J, there exist p , q j with π ¯ 1 ( π ( p ) ) , π 1 ( π ¯ ( q ) ) { 1 , , j } . By (1.3), the coefficients of X π ( p ) are equal: i = p N λ i i = i = π ¯ 1 ( π ( p ) ) N μ i i . The same holds for X π ( q ) : i = q N μ i i = i = π 1 ( π ¯ ( q ) ) N λ i i . Put together
i = j + 1 N μ i i i = q N μ i i = i = π 1 ( π ¯ ( q ) ) N λ i i i = j + 1 N λ i i i = p N λ i i = i = π ¯ 1 ( π ( p ) ) N μ i i i = j + 1 N μ i i ,

which is only possible if μ j = λ j = 0 since p , q j . Furthermore, if C π C π ¯ is of dimension N, by (1.5) it follows that π = π ¯ . This shows (iii).

Further, for B s = conv ( X 1 , , X s ) , the elements C π B s of dimension s are exactly the ones with { π ( i ) : i = 1 , , s } = { 1 , , s } . To this end, let C π B s be of dimension s. This means there exists an element Y in this intersection such that Y = i = 1 N λ i X i with λ i > 0 for all i = 1 , , s and λ i = 0 for i > s . As an element of C π , this Y has a representation of the form Y = j = 1 N ( k = j N μ k k ) X π ( j ) for k = 1 N μ k = 1 and μ k L + 0 for every k = 1 , , N . Suppose now that there exists some j 0 s with π ( j 0 ) > s . Then due to λ π ( j 0 ) = 0 and the uniqueness of the coefficients (cf. (1.3)) in a conditional simplex, it holds that k = j 0 N μ k k = 0 and within k = j N μ k k = 0 for all j j 0 . This means Y = j = 1 j 0 1 ( k = j N μ k k ) X π ( j ) and hence Y is the convex combination of j 0 1 elements with j 0 1 < s . This contradicts the property that λ i > 0 for s elements. Therefore, ( C π B s ) π is exactly the barycentric subdivision of B s , which has been shown to fulfill the properties (i)-(iii). □

Subdividing a conditional simplex S = conv ( X 1 , , X N ) barycentrically, we obtain ( C π ) π S N . Dividing every C π barycentrically results in a new collection of conditional simplexes and we call this the two-fold barycentric subdivision of S . Inductively, we can subdivide every conditional simplex of the ( m 1 ) th step barycentrically and call the resulting collection of conditional simplexes the m-fold barycentric subdivision of S and denote it by S m . Further, we define ext ( S m ) = σ ( { ext ( C ) : C S m } ) to be the σ-stable hull of all extremal points of the conditional simplexes of the m-fold barycentric subdivision of S . Notice that this is the σ-stable hull of only finitely many elements, since there are only finitely many simplexes in the subdivision, each of which is the convex hull of N elements.

Remark 1.12 Consider an arbitrary C π = conv ( Y 1 π , , Y N π ) , π S N in the barycentric subdivision of a conditional simplex S . Then it holds that
diam ( C π ) = ess sup i , j = 1 , , N Y i π Y j π N 1 N diam ( S ) .

Since this holds for any π S N , it follows that the diameter of S m , which is an arbitrary conditional simplex of the m-fold barycentric subdivision of S , fulfills diam ( S m ) ( N 1 N ) m diam ( S ) . Since diam ( S ) < and ( N 1 N ) m 0 , for m , it follows that diam ( S m ) 0 for m for every sequence ( S m ) m N .

2 Brouwer fixed point theorem for conditional simplexes

Definition 2.1 Let S = conv ( X 1 , , X N ) be a conditional simplex, m-fold barycentrically subdivided in S m . A local function ϕ : ext ( S m ) { 1 , , N } ( A ) is called a labeling function of S . For fixed X 1 , , X N ext ( S ) with S = conv ( X 1 , , X N ) , the labeling function is called proper if for any Y ext ( S m ) it holds that
P ( { ω : ϕ ( Y ) ( ω ) = i , λ i ( ω ) = 0 } ) = 0
for i = 1 , , N , where Y = i = 1 N λ i X i . A conditional simplex C = conv ( Y 1 , , Y N ) , with C S , with Y j ext ( S m ) , j = 1 , , N , is said to be completely labeled by ϕ if ϕ is a proper labeling function of S and
P ( { ω : there exists  j { 1 , , N } , ϕ ( Y j ) ( ω ) = i } ) = 1

for all i { 1 , , N } .

Lemma 2.2 Let S = conv ( X 1 , , X N ) be a conditional simplex and f : S S be a local function. Let ϕ : ext ( S m ) { 0 , , N } ( A ) be a local function such that for every X ext ( S m ) it holds that
  1. (i)

    P ( { ω : ϕ ( X ) ( ω ) = i ; λ i ( ω ) = 0 or μ i ( ω ) > λ i ( ω ) } ) = 0 for all i = 1 , , N ,

     
  2. (ii)

    P ( { ω : ϕ ( X ) ( ω ) = 0 , i { 1 , , N } , λ i ( ω ) > 0 , λ i ( ω ) μ i ( ω ) } ) = 0 ,

     

where ( λ i ) i = 1 , , N and ( μ i ) i = 1 , , N are determined by X = i = 1 N λ i X i and f ( X ) = i = 1 N μ i X i . Then ϕ is a proper labeling function.

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

Proof First we show that ϕ is a labeling function. Since ϕ is local, we just have to prove that ϕ actually maps into { 1 , , N } . Due to (ii), we have to show that
P ( { ω : there exists  i { 1 , , N } , λ i ( ω ) μ i ( ω ) , λ i ( ω ) > 0 } ) = 1 .

Assume, to the contrary, that μ i > λ i on A A + for all λ i with λ i > 0 on A. Then it holds that 1 = i = 1 N λ i 1 { λ i > 0 } < i = 1 N μ i 1 { μ i > 0 } = 1 on A, which yields a contradiction. Thus, ϕ is a labeling function. Moreover, due to (i), it holds in particular that P ( { ω : ϕ ( X ) ( ω ) = i , λ i ( ω ) = 0 } ) = 0 , which shows that ϕ is proper.

To prove the existence for X ext ( S m ) with X = i = 1 N λ i X i , f ( X ) = i = 1 N μ i , let B i : = { ω : λ i ( ω ) > 0 } { ω : λ i ( ω ) μ i ( ω ) } , i = 1 , , N . Then we define the function ϕ at X as { ω : ϕ ( X ) ( ω ) = i } = B i ( k = 1 i 1 B k ) , i = 1 , , N . It has been shown that ϕ maps to { 1 , , N } ( A ) and is proper. It remains to show that ϕ is local. To this end, consider X = j N 1 A j X j , where X j = i = 1 N λ i j X i and f ( X j ) = i = 1 N μ i j X i . Due to uniqueness of the coefficients in a conditional simplex, it holds that λ i = j N 1 A j λ i j , and due to locality of f, it follows that μ i = j N 1 A j μ i j . Therefore it holds that B i = j N ( { ω : λ i j ( ω ) > 0 } { ω : λ i j ( ω ) μ i j ( ω ) } A j ) = j N ( B i j A j ) . Hence, ϕ ( X ) = i on B i ( k = 1 i 1 B k ) = [ j N ( B i j A j ) ] [ k = 1 i 1 ( j N B k j A j ) ] = j N [ ( B i j k = 1 i 1 B k j ) A j ] . On the other hand, we see that j N 1 A j ϕ ( X j ) is i on any A j { ω : ϕ ( X j ) ( ω ) = i } , hence it is i on j N ( B i j k = 1 i 1 B k j ) A j . Thus, j N 1 A j ϕ ( X j ) = ϕ ( j N 1 A j X j ) , which shows that ϕ is local. □

The reason to demand locality of a labeling function is exactly because we want to label by the function ϕ mentioned in the existence proof of Lemma 2.2 and hence keep local information with it. For example, consider a conditional simplex S = conv ( X 1 , X 2 , X 3 , X 4 ) and Ω = { ω 1 , ω 2 } . Let Y ext ( S ) be given by Y = 1 3 i = 1 3 X i . Now consider a function f on S such that
f ( Y ) ( ω 1 ) = 1 4 X 1 ( ω 1 ) + 3 4 X 3 ( ω 1 ) ; f ( Y ) ( ω 2 ) = 2 5 X 1 ( ω 2 ) + 2 5 X 2 ( ω 2 ) + 1 5 X 4 ( ω 2 ) .

If we label Y by the rule explained in Lemma 2.2, ϕ takes the values ϕ ( Y ) ( ω 1 ) { 1 , 2 } and ϕ ( Y ) ( ω 2 ) = 3 . Therefore, we can really distinguish on which sets λ i μ i . Yet, using a deterministic labeling of Y, we would lose this information.

Theorem 2.3 Let S = conv ( X 1 , , X N ) be a conditional simplex in ( L 0 ) 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 ( C π ) π S N of S and a proper labeling function ϕ on ext ( S ) . First, we show that we can find a completely labeled conditional simplex in S . By induction on the dimension of S = conv ( X 1 , , X N ) , we show that there exists a partition ( A k ) k = 1 , , K such that on any A k there is an odd number of completely labeled C π . The case N = 1 is clear since a point can be labeled with the constant index 1 only.

Suppose that the case N 1 is proven. Since the number of Y i π of the barycentric subdivision is finite and ϕ can only take finitely many values, it holds for all V ( Y i π ) i = 1 , , N , π S N that there exists a partition ( A k V ) k = 1 , , K , K < , where ϕ ( V ) is constant on any A k V . Therefore, we find a partition ( A k ) k = 1 , , K such that ϕ ( V ) on A k is constant for all V and A k . Fix A k now.

In the following, we denote by C π b those conditional simplexes for which C π b B N 1 are N 1 -dimensional (cf. Lemma 1.11(iv)), therefore π b ( N ) = N . Further we denote by C π c these conditional simplexes which are not of the type C π b , that is, π c ( N ) N . If we use C π , we mean a conditional simplex of arbitrary type. We define:

  • C ( C π ) π S N to be the set of C π which are completely labeled on A k .

  • A ( C π ) π S N to be the set of P-almost completely labeled C π , that is,
    { ϕ ( Y k π ) , k { 1 , , N } } = { 1 , , N 1 } on  A k .
  • E π to be the set of intersections ( C π C π l ) π l S N which are N 1 -dimensional and completely labeled on A k .b

  • B π to be the set of intersections C π B N 1 which are completely labeled on A k .

It holds that E π B π = and hence | E π B π | = | E π | + | B π | . Since C π c B N 1 is at most N 2 -dimensional, it holds that B π c = and hence | B π c | = 0 . Moreover, we know that C π C π l is N 1 -dimensional on A k if and only if this holds on the whole Ω (cf. Lemma 1.11(iii)) and C π b B N 1 on A k if and only if this also holds on the whole Ω (cf. Lemma 1.11(iv)). So, it does not play any role if we look at these sets which are intersections on A k or on Ω since they are exactly the same sets.

If C π c C , then | E π c | = 1 and if C π b C , then | E π b B π b | = 1 . If C π c A , then | E π c | = 2 and if C π b A , then | E π b B π b | = 2 . Therefore it holds that π S N | E π B π | = | C | + 2 | A | .

If we pick E π E π , we know that there always exists exactly one other π l such that E π E π l (Lemma 1.11(iii)). Therefore π S N | E π | is even. Moreover, ( C π b B N 1 ) π b subdivides B N 1 barycentrically, and hence we can apply the inductive hypothesis (on ext ( C π b B N 1 ) ). Indeed, the set B N 1 is a σ-stable set, so if it is partitioned by the labeling function into ( A k ) k = 1 , , K , we know that B N 1 ( S ) = k = 1 K 1 A k B N 1 ( 1 A k S ) and by Lemma 1.11(iv) we can apply the induction hypothesis also to every A k , k = 1 , , K . Thus, the number of completely labeled conditional simplexes is odd on a partition of Ω, but since ϕ is constant on A k , it also has to be odd there. This means that π b | B π b | has to be odd. Hence, we also have that π | E π B π | 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 any A k a completely labeled  C π k .

We define S 1 = k = 1 K 1 A k C π k which by Remark 1.9 is indeed a conditional simplex. Due to σ-stability of S , it holds that S 1 S . By Remark 1.12, S 1 has a diameter which is less than N 1 N diam ( S ) and since ϕ is local, S 1 is completely labeled on the whole Ω.

The same argumentation holds for every m-fold barycentric subdivision S m of S , m N , that is, there exists a completely labeled conditional simplex in every m-fold barycentrically subdivided conditional simplex which is properly labeled. Henceforth, subdividing S m-fold barycentrically and labeling it by ϕ m : ext ( S m ) { 1 , , N } ( A ) , which is a labeling function as in Lemma 2.2, we always obtain a completely labeled conditional simplex S m + 1 S for m N . Moreover, since S 1 is completely labeled, it holds S 1 = k = 1 K 1 A k C π k as above, where C π k is completely labeled on A k . This means C π k = conv ( Y 1 k , , Y N k ) with ϕ ( Y j k ) = j on A k for every j = 1 , , N . Defining V j 1 = k = 1 K 1 A k Y j k for every j = 1 , , N yields P ( { ω : ϕ ( V j 1 ) ( ω ) = j } ) = 1 for every j = 1 , , N and S 1 = conv ( V 1 1 , , V N 1 ) . The same holds for any m N and so that we can write S m = conv ( V 1 m , , V N m ) with P ( { ω : ϕ m 1 ( V j m ) ( ω ) = j } ) = 1 for every j = 1 , , N .

Now, ( V 1 m ) m N is a sequence in the sequentially closed, L 0 -bounded set S , so that by [[1], Corollary 3.9], there exists Y S and a sequence ( M m ) m N in N ( A ) such that M m + 1 > M m for all m N and lim m V 1 M m = Y P-almost surely. For M m N ( A ) , V 1 M m is defined as n N 1 { M m = n } V 1 n . This means an element with index M m , for some m N , equals V 1 n on A n , n N , where the sets A n are determined by M m via A n = { ω : M m ( ω ) = n } , n N . Furthermore, as m goes to ∞, diam ( S m ) is converging to zero P-almost surely, and therefore it also follows that lim m V k M m = Y P-almost surely for every k = 1 , , N . Indeed, it holds that | V k m Y | diam ( S m ) + | V 1 m Y | for every k = 1 , , N and m N , so we can use the sequence ( M m ) m N for every k = 1 , , N .

Let Y = l = 1 N α l X l and f ( Y ) = l = 1 N β l X l as well as V k m = l = 1 N λ l m , k X l and f ( V k m ) = l = 1 N μ l m , k X l for m N . As f is local, it holds that f ( V 1 M m ) = n N 1 { M m = n } f ( V 1 n ) . By sequential continuity of f, it follows that lim m f ( V k M n ) = f ( Y ) P-almost surely for every k = 1 , , N . In particular, lim m λ l M m , l = α l and lim m μ l M m , l = β l P-almost surely for every l = 1 , , N . However, by construction, ϕ m 1 ( V l m ) = l for every l = 1 , , N , and from the choice of ϕ m 1 , it follows that λ l m , l μ l m , l P-almost surely for every l = 1 , , N and m N . Hence, α l = lim m λ l M m , l lim m μ l M m , l = β l P-almost surely for every l = 1 , , N . This is possible only if α l = β l P-almost surely for every l = 1 , , N , showing that f ( Y ) = Y . □

3 Applications

3.1 Fixed point theorem for sequentially closed and bounded sets in ( L 0 ) d

Proposition 3.1 Let K be an L 0 -convex, sequentially closed and bounded subset of ( L 0 ) d , and let f : K K be a local, sequentially continuous function. Then f has a fixed point.

Proof Since K is bounded, there exists a conditional simplex S such that K S . Now define the function h : S K by
h ( X ) = { X , if  X K , arg min { X Y : Y K } , else .

This means, that h is the identity function on K and the projection on K for the elements in S K . Due to [[1], Corollary 4.5] this minimum exists and is unique. Therefore h is well defined.

We can characterize h by
Y = h ( X ) X Y , Z Y 0 for all  Z K .
(3.1)
Indeed, let X Y , Z Y 0 for all Z K . Then
X Z 2 = ( X Y ) + ( Y Z ) = X Y 2 + 2 X Y , Y Z + Y Z 2 X Y 2 ,
which shows the minimizing property of h. On the other hand, let Y = h ( X ) . Since K is L 0 -convex, λ Z + ( 1 λ ) Y K for any λ ( 0 , 1 ] ( A ) and Z K . By a standard calculation,
X ( λ Z + ( 1 λ ) Y ) 2 X Y 2
yields 0 2 λ X , Y + ( 2 λ λ 2 ) Y , Y + 2 λ X , Z λ 2 Z 2 2 λ ( 1 λ ) Z , Y . Dividing by λ > 0 and letting λ 0 afterwards yields
0 2 X , Y + 2 Y , Y + 2 X , Z 2 Z , Y = 2 X Y , Z Y ,

which is the desired claim.

Furthermore, for any X , 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 2 = h ( X ) h ( Y ) 2 + c 2 + 2 c , h ( X ) h ( Y ) .
(3.2)
Since
c , h ( X ) h ( Y ) = X h ( X ) , h ( Y ) h ( X ) Y h ( Y ) , h ( X ) h ( Y ) ,

by (3.1), it follows that c , h ( X ) h ( Y ) 0 . Therefore, X Y 2 h ( X ) h ( Y ) 2 by (3.2). This shows that h is sequentially continuous.

The function f h is a sequentially continuous function mapping from S to K S . Hence, there exists a fixed point f h ( Z ) = Z . Since f h maps into K , this Z has to be in  K . But then we know h ( Z ) = Z and therefore f ( Z ) = Z , which ends the proof. □

Remark 3.2 In Drapeau et al. [9] 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 ( L 0 ) d . In that context we can formulate the conditional Brouwer fixed point theorem as follows. A sequentially continuous function f : K K such that K is a conditionally compact and L 0 -convex subset of ( L 0 ) d has a fixed point.

3.2 Applications in conditional analysis on ( L 0 ) d

Working in R d , 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 0 ) d , we will show that the conditional Brouwer fixed point theorem has several implications as well.

Define the unit ball in ( L 0 ) d by B ( d ) = { X ( L 0 ) d : X 1 } . Then, by the former theorem, any local, sequentially continuous function f : B ( d ) B ( d ) has a fixed point. The unit sphere S ( d 1 ) is defined as S ( d 1 ) = { X ( L 0 ) d : X = 1 } .

Definition 3.3 Let X and Y be subsets of ( L 0 ) d . An L 0 -homotopy of two local, sequentially continuous functions f , g : X Y is a jointly local, sequentially continuous function H : X × [ 0 , 1 ] ( A ) Y such that H ( X , 0 ) = f ( X ) and H ( X , 1 ) = g ( X ) . Jointly local means H ( j N 1 A j X j , j N 1 A j t j ) = j N 1 A j H ( X j , t j ) for any partition ( A j ) j N , ( X j ) j N in X and ( t j ) j N in [ 0 , 1 ] ( A ) . Sequential continuity of H is therefore H ( X n , t n ) H ( X , t ) whenever X n X and t n t both P-almost surely for X n , X X and t n , t [ 0 , 1 ] ( A ) .

Lemma 3.4 The identity function of the sphere is not L 0 -homotopic to a constant function.

The proof is a consequence of the following lemma.

Lemma 3.5 There does not exist a local, sequentially continuous function f : B ( d ) S ( d 1 ) which is the identity on S ( d 1 ) .

Proof Suppose that there is this local, sequentially continuous function f. Define the function g : S ( d 1 ) S ( d 1 ) by g ( X ) = X . Then the composition g f : B ( d ) B ( d ) , which actually maps to S ( d 1 ) , is local and sequentially continuous. Therefore, this has a fixed point Y which has to be in S ( d 1 ) since this is the image of g f . But we know f ( Y ) = Y and g ( Y ) = Y and hence g f ( Y ) = Y . Therefore, Y cannot be a fixed point (since 0 S ( d 1 ) ), which is a contradiction. □

It directly follows that the identity on the sphere is not L 0 -homotopic to a constant function. In the case d = 1 , we get the following result which is the L 0 -module version of an intermediate value theorem.

Lemma 3.6 Let X , X ¯ L 0 with X X ¯ and [ X , X ¯ ] : = { Z L 0 : X Z X ¯ } . Let further f : [ X , X ¯ ] L 0 be a local, sequentially continuous function and A : = { ω : f ( X ) ( ω ) f ( X ¯ ) ( ω ) } . If Y is in [ 1 A f ( X ) + 1 A c f ( X ¯ ) , 1 A f ( X ¯ ) + 1 A c f ( X ) ] , then there exists Y ¯ [ X , X ¯ ] with f ( Y ¯ ) = Y .

Proof Since f is local, it is sufficient to prove the case for f ( X ) f ( X ¯ ) which is A = Ω . For the general case, we would consider A and A c separately, obtain 1 A f ( Y ¯ 1 ) = 1 A Y , 1 A c f ( Y ¯ 2 ) = 1 A c Y and by locality we have f ( 1 A Y ¯ 1 + 1 A c Y ¯ 2 ) = Y . So, suppose that Y is in [ f ( X ) , f ( X ¯ ) ] in the rest of the proof.

Let first f ( X ) < Y < f ( X ¯ ) . Define the function g : [ X , X ¯ ] [ X , X ¯ ] by
g ( V ) : = p ( V f ( V ) + Y ) with p ( Z ) = 1 { Z X ¯ X } X + 1 { X Z X ¯ } Z + 1 { X ¯ Z } X ¯ .

Notice that as a sum, product, and composition of local, sequentially continuous functions, g is so as well. Hence, g has a fixed point Y ¯ . If Y ¯ = X , it must hold that X f ( X ) + Y X , which means Y f ( X ) , which is a contradiction. If Y ¯ = X ¯ , it follows that f ( X ¯ ) Y , which is also a contradiction. Hence, Y ¯ = Y ¯ f ( Y ¯ ) + Y , which means f ( Y ¯ ) = Y .

If Y = f ( X ) on B and Y = f ( X ¯ ) on C, then f ( X ) < Y < f ( X ¯ ) on D : = ( B C ) c . Then we find Y ¯ such that f ( Y ¯ ) = Y on D. In total f ( 1 B X + 1 C B X ¯ + 1 D Y ¯ ) = 1 B f ( X ) + 1 C B f ( X ¯ ) + 1 D f ( Y ¯ ) = Y . This shows the claim for general Y in [ f ( X ) , f ( X ¯ ) ] . □

Endnotes

Let B π : = { ω : λ π ( 1 ) ( ω ) λ π ( 2 ) ( ω ) λ π ( N ) ( ω ) } , π S N . This finite collection of measurable sets fulfills P ( π S N B π ) = 1 . We can construct a partition ( A n ) n = 1 , , M such that A n B π n for some π n S N and for all n = 1 , , M . Such a partition fulfills the required property.

That is bearing exactly the label 1 , , N 1 on A k .

Declarations

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.

Authors’ Affiliations

(1)
Humboldt-Universität zu Berlin
(2)
Universität Konstanz

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© Drapeau et al.; licensee Springer. 2013

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