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On approximation of asymmetric separators of the n-cube

Abstract

A new combinatorial result intertwined with the Brouwer fixed point theorem for the n-cube is given. This result can be used for any map (f1, ..., fn): [0, 1]n → [0, 1]n to approximate the components of the set {(x1, . . . , x n ) [0, 1]n : f i (x1, . . . , x n ) = x i } that separate the n-cube between the i th opposite faces. Equivalently, for maps g : [0, 1]n such that g(x)g(y) ≤ 0 for any x {0} × [0, 1]n-1and y {1} × [0, 1]n-1, one can use the algorithm to approximate the components of g-1(0) that separate [0, 1]n between {0} × [0, 1]n-1and {1} × [0, 1]n-1. The methods are based on an earlier result of P. Minc and the present authors and relate to results of several other authors such as Jayawant and Wong, Kulpa and Turzański, and Gale.

Mathematics Subject Classification (2000): Primary 54H25; 54-04; Secondary 55M20; 54F55.

1. Introduction

In [1] Minc and the present authors described combinatorial methods that allow approximation of connected symmetric separators of the n-sphere and n-cube. The symmetric separators arise in the context of the Borsuk-Ulam antipodal theorem and a theorem of Dyson for the 2-sphere [2, 3]. The purpose of the present paper is to show how the results from [1] can be extended to the setting of asymmetric separators and the Brouwer fixed point theorem for the n-cube. The classic result of L.E.J. Brouwer says that the n-dimensional cube In = [0, 1]n has the fixed point property; that is, for any mapa f : InIn, there is x In such that f(x) = x. There are many important applications of the Brouwer's theorem such as, for example, those concerning existence of solutions for differential equations [4], or equilibrium strategies in multi-person games relating to market problems in economics [5]. This is why computability of fixed points became an important theme in the fixed point theory. The first fixed point algorithm was given by Scarf [6]. Soon after, other were given by Eaves [7] and Todd [8] (see, for example, [9, 10] for a comprehensive treatment of this subject with applications). There are also several combinatorial equivalents of Brouwer's theorem. The best known is probably Sperner's lemma [11] on coloring vertices of a barycentric subdivision of an n-simplex. Some other transfer the fixed point problem to the scenario of board games, such as Hex [12] or Chess [13].

In the present paper, in Theorem 3.1, we formulate yet another combinatorial result that implies the Brouwer fixed point theorem. Its baby version can be formulated as follows.

Theorem 1.1. Suppose f : V is a function defined on the set of vertices V of a triangulation T of In. Suppose in addition that f(v1)f(v2) ≤ 0 for any vertices v1 {0} × In-1and v2 {1} × In-1. Then, there is a subcollection S T of simplices of dimension n such that

  1. 1.

    for every simplex σ S there is an edge [v, u] such that f (v) f (u) ≤ 0;

  2. 2.

    S separates In between {0} × In-1and {1} × In-1.

The above theorem implies the Brouwer fixed point theorem in the following way. If (f1, . . . , f n ): InInis a map and X is a polyhedral complex with In as its underlying space, then each g i (x1, . . . , x n ) = f i (x1, . . . , x n ) -x i satisfies the assumptions of Theorem 1.1 and one can find C i , an approximation of a component of g i - 1 ( 0 ) , that separates In between the i th opposite (n - 1)-dimensional faces. By Eilenberg-Otto theorem (see [14]) i = 1 n C i is nonempty and approximates a fixed point of f.

We give a stronger (but at the same time more technical) version of the above result in Theorem 3.1, and in Section 4, we show how along with Theorem 4.1 it can be used to approximate a connected separating component of the set of zeros of an arbitrary map f : In, which assumes opposite signs on some two opposite (n - 1)-faces of In. The case when n = 2 was already considered in [15]. The methods used in the proof of Theorem 3.1 are based on those introduced in [1] where, in connection with the Borsuk-Ulam antipodal theorem, it was shown how to approximate a connected separator of the n-sphere Sn(or In), invariant under the antipodal map. Any such separator was corresponding to a component of f-1(0), with f : Sn (or f : In) an odd map (related combinatorial results can be found in [2, 16]). However, the methods of [1] were dealing only with symmetric separators and are insufficient in the case of arbitrary separators. First, unlike in the case of symmetric separators and odd maps, if a map f : In→ R satisfies the condition f({1} × In-1) [0, ∞) and f({0} × In-1) (-∞, 0] for some i = 1, . . . , n, there may be no unique connected separator of Inin f-1(0). Clearly, f-1(0) may consist of several disjoint separating components, none of which needs to be symmetric. Second, the algorithms in [1] were making use of the fact that the symmetric component of f-1(0) is the separating omponent, when f is odd. Therefore, if a subcollection of the triangulation approximated a component of f-1(0) and, at the same time, was symmetric, this was sufficient to determine that it separated Sn(or In). This is why one is forced to develop new combinatorial criteria for arbitrary separators in In. In Section 4 we furnish such a computer implementable criterion that allows isolating those subcollections of the triangulation, approximating a component of f-1(0), that separate In from those that do not.

2. Preliminaries

For a collection of sets K, by K * we will denote the union of all its elements. π i : [0, 1]n → [0, 1] will denote the projection onto the i th coordinate. I i + and I i - will denote the i th opposite (n - 1)-dimensional faces of In, that is I i + = π i - 1 ( 1 ) and I i - = π i - 1 ( 0 ) . C separates In (or is a separator of In) between I i + and I i - if for any x I i + \C,y I i - \C, there are U, V , distinct components of In\C, such that x U and y V. A map g: X * is piecewise linear if given {σ j : j = 1, . . . , N}, a triangulation of X * , for every j the restriction of g to the simplex σ j is linear, that is g i = 1 k λ i a i = i = 1 k λ i g ( a i ) where a1, . . . , a k are the vertices spanning σ j and λ i ≥ 0 with i = 1 k λ i = 1 (see [17]).

We will heavily rely on the following inductive procedure introduced by Minc and the two authors in [1]. Let X be a polyhedral complex such that X * = [ 0 , 1 ] n . Let V ( X ) and E ( X ) denote the collections of vertices and edges, respectively. Suppose f:V ( X ) is a function. Let E f be the collection of those edges e= u , v E ( X ) that f (u) f (v) ≤ 0. Let PX be the collection of polytopes in X of dimension n. For any e E f , C ( e ) is defined by induction.

  • Let C 0 ( e ) be the collection of those PP that contain e.

  • Suppose C i - 1 ( e ) has been defined. Define C i ( e ) to be the collection of those PP such that the intersection P C i - 1 ( e ) * contains an edge from E f or a vertex from f-1 (0).

Clearly, C i - 1 ( e ) C i ( e ) and there is an integer q ≥ 0 such that C q ( e ) = C q + 1 ( e ) . For the first such number q (e) set C ( e ) = C q ( e ) ( e ) . Note that e C ( e ) * , and C ( e ) * is connected.

3. Combinatorial theorem on separators of Inbetween opposite faces

Let χ be a polyhedral complex such that X * = [ 0 , 1 ] n . Note that X can be subdivided to give a triangulation of In, without introducing new vertices [18], and consequently, every function f:V ( X ) has a piecewise linear extension g : In . The following result is of purely combinatorial nature.

Theorem 3.1. Suppose that f:V ( X ) is a function satisfying

f ( V ( X ) I i + ) [ 0 , + ) , f ( V ( X ) I i - ) ( - , 0 ]
(3.1)

for some i {1, . . . ,n}. Let L be a subcollection of E ( X ) such that L * I j ε for some j ≠ i andε {+, -}, and L * is an arc with endpoints in I i + and I i - . Then, there is an edge dE ( X ) L such that d E f and

  1. 1.

    each CC ( d ) contains an edge from E f ,

  2. 2.

    C ( d ) * separates In between I i + and I i - and

  3. 3.

    for any other such d L with C ( d ) * satisfying ( 1)-( 2), either C ( d ) C ( d ) = or C ( d ) =  C ( d ) .

Proof. Without loss we can assume that i = 1. Let g : In be a piecewise linear extension of f. Then, g is continuous and g(v) = f(v) for any vV ( X ) .

Claim 3.1.1. If KX, then g(r) = 0 for some r K if and only if there is an edge [ w , v ] E ( X ) such that w,vKV ( X ) and f (w) f (v) ≤ 0.

Proof of claim 3.1.1. First suppose KX is such that there are vertices w,vKV ( X ) , and f (w) f (v) ≤ 0. Then, either f(w)f(v) = 0 or f(w)f(v) < 0. In the first case, clearly g(w) = f(w) = 0 or g(v) = f(v) = 0. Otherwise there must be r [u, v] such that g(r) = 0. For the converse, suppose g(r) = 0 for some r K. Then, r = i = 1 k λ i a i , where a o , . . . , a k are vertices of K spanning a simplex σ K, and λ i ≥ 0 with i = 1 k λ i =1. Therefore, 0=g ( r ) = i = 1 k λ i g ( a i ) = i = 1 k λ i f ( a i ) . Clearly, there is l such that f(a l ) = 0, or there are a j , a t such that f(a j )f(a t ) < 0.   □

Claim 3.1.2. g ( I i + ) [ 0 , + ) ,g ( I i - ) ( - , 0 ] .

Proof of claim 3.1.2. Similarly to the proof of claim 3.1.1, this follows from the fact that if a o , . . . , a k spans a simplex σ and g (a i ) ≥ 0 (g(a i ) ≤ 0) for every i, then i = 1 k λ i g ( a i ) 0 ( i = 1 k λ i g ( a i ) 0 ) for λ i ≥ 0. Consequently, g(σ) [0,+∞) (g(σ) (-∞, 0]) for any such σ.   □

Now, consider the following decomposition of the n-cube.

Z = {x In : g(x) = 0}, N = {x In : g(x) < 0}, P = {x In : g(x) > 0}. Clearly Z separates In between P and N. Let Z1, . . . , Z p be the components of Z. It is well known that if X is a connected, locally connected and unicoherent space then any closed set separating X contains a connected subset separating X ([[19], p. 195], cf. [[20], p. 429, Theorem 1.(vi)]). Since Z is closed and separates In, by unicoherence of In, there must be q such that Z q separates In between N and P. Consequently, Z q separates In between I i + and I i - by claim 3.1.2.

Consider S a subcollection of K such that KS if, it has nonempty intersection with Z q , that is S= { K X : K Z q } . Clearly Z q S * and therefore S * separates In between I i + and I i - .

Now, let L be a subcollection of E ( X ) such that L * I j ε for some j ≠ 1 and ε {+, -}, and L * is an arc with endpoints a I 1 + and b I 1 - , that is L * = [ a , b ] . Since Z q ∩ Bd (In) separates Bd (In) between I i + and I i - , we conclude there is z [a, b] ∩ Z q . Additionally, there is dL such that z d. By claim 3.1.1 d E f L, and since dZ q therefore dS.

Claim 3.1.3. SC ( d ) .

Proof of claim 3.1.3. Let LS be such that d L. Clearly L C 0 ( d ) C ( d ) . Heading toward a contradiction suppose S\C ( d ) . Consider a partition of S into the following two sets

S 1 = { T S : T C ( d ) } , S 2 = { T S : T C ( d ) } .

By definition of C ( d ) , for any T S 2 and for any T ̃ S 1 , we must have that whenever sE ( X ) and sT T ̃ then s E f . Otherwise T would be in S 1 . Therefore, ( T T ̃ ) Z q =, by claim 3.1.1. Consequently, there is a partition of Z q into two disjoint sets Z q S 1 * and Z q S 2 * . Since both are closed, we obtain a contradiction with connectedness of Z q .   □

Now, property (1) is an immediate consequence of the definition of C ( d ) . Since S * * C ( d ) , (2) easily follows from the fact that S * separates In between I i + and I i - . Now, suppose d L is another edge with C ( d ) * satisfying (1)-(2). If C ( d ) C ( d ) , then there is K such that K C j ( d ) and K C p ( d ) for some j and p. Consequently, i = 0 j C i ( d ) i = p q ( d ) C i ( d ) and i = j q ( d ) C i ( d ) i = p q ( d ) C i ( d ) , by definition of C ( d ) . Clearly C ( d ) C ( d ) . Similarly C ( d ) C ( d ) . That justifies (3) and completes the proof.   □

4. Algorithm approximating connected separators of In

Suppose K is a partition of In into kn congruent n-cubes, all with side length equal to 1 k . In this section we shall furnish a computer implementable criterion for the union of a subcollection of K to separate In between some two opposite faces. Suppose SK and we want to determine if S * separates In between I i + and I i - .

Let KK\S be an n-cube. G is a j-face of K if dim (G) = j and G = KL for some K, LK. We will define Comp ( K , S ) by induction. Let Comp1 ( K , S ) consists of K and all those cubes L in K\S such that KL is an (n - 1)-face. Suppose Comp p ( K , S ) has already been defined and let Compp+1 ( K , S ) consists of all cubes in Comp p ( K , S ) , and all those cubes R in K\S for which there is a cube L Comp p ( K , S ) such that LR is an (n - 1)-face. Since K consists of only finite number of cubes Comp q ( K , S ) = Compq+1 ( K , S ) for some natural number q. Let q(K) be the first such number and let Comp ( K , S ) = Com p q ( K ) ( K , S ) .

Theorem 4.1. S * separates In between I i + and I i - iff

Comp  ( K , S ) * I i = f o r e v e r y K K s u c h t h a t K I i + .
(4.1)

Proof. If the condition (4.1) is not satisfied, then clearly S * does not separate In between I i + and I i - . Namely, Comp ( K , S ) * for some K contains a connected set, disjoint with S * , intersecting both I i + and I i - in a nonempty set. For the converse, by contradiction suppose that the condition (4.1) is satisfied but S * does not separate In between I i + and I i - . Let A be a connected component of I n \ S * intersecting both I i + and I i - in a nonempty set. Let R be a subcollection of K\S such that R * is connected and A R * . Without loss of generality, we can assume that R is a minimal such collection. We will obtain a contradiction showing, by induction, that for any two cubes in R if their intersection is an m-face, then m {0, . . . , n - 2}. Suppose K, LR are two cubes such that KL but L Comp ( K , S ) . Since KL must be an m-face, for some m < n, we must have that KL is an m-face with m < n - 1. Suppose m = n - 2, then there are exactly 22 - 2 other cubes sharing this m-face. Let T be any of those two cubes. Then, TK and TL are (n - 1)-faces, KL T and T must be in S. A contradiction with the fact that (KL) ∩ A and therefore mn - 2. Suppose we have already proved that m < n - i. We shall show that mn - (i + 1). Suppose otherwise, that is KL is an n - (i + 1)-face, for some K, LR. Then, there are 2i+1- 2 other cubes having this n - (i + 1)-face in common. Let T be one of them such that TK is an (n - 1)-face. Then, TL is an (n - i)-face and KL T. Since (KL) T and (KL) ∩ A, therefore AT and TS. Consequently, T Comp ( K , S ) with TL an (n - i)-face, which leads to a contradiction by an inductive step.

It follows that for any two K, LR we have L Comp ( K , S ) . Consequently, R= Comp ( K , S ) for some K such that K I i + and Comp ( K , S ) * I i - . A contradiction that completes the proof.   □

A collection of cubes in K and the collection of the faces of all dimensions of cubes in K forms a polyhedral complex with K as its generating collection. Denote this complex by X.

Suppose f:V ( X ) is such that f (v) ≥ 0 for each vV ( X ) I 1 + and f (v) ≤ 0 for each vV ( X ) I i - . We will make use of Theorems 3.1 and 4.1 to obtain an algorithm finding C ( d ) K, for all d E f , such that the following is true

  1. 1.

    each CC ( d ) contains an edge from E f ,

  2. 2.

    C ( d ) * is connected, and

  3. 3.

    C ( d ) * separates In between x I 1 + and y I 1 - for each x,y I n \C ( d ) * .

Set L= { [ i k , i + 1 k ] × { 0 } × . . . × { 0 } : i = 0 , . . . , k - 1 } , and notice that L * is a segment joining I i + and I i - . Therefore, C ( d ) will be the desired collection satisfying (1) - (3) for some dL.

Algorithm (outline)

Step 1. Add all elements of E f L to List A.

Step 2. Repeat Step 3-Step 11 until List A is empty.

Step 3. Pick an edge d from List A.

Step 4. Generate C ( d ) . Remove d from List A.

Step 5. Add all elements KK such that K I i + to List B.

Step 6. Repeat Step 7-Step 9 until List B is empty.

Step 7. Pick a cube K from List B.

Step 8. Generate Comp (K, C ( d ) ). Remove K from List B.

Step 9. If there is L Comp (K, C ( d ) ) such that L I i - then go back to Step 3.

Otherwise, go back to Step 7.

Step 10. List all elements from C ( d ) (C ( d ) * is a separator).

Step 11. Go back to Step 3.

Endnote

aBy a map we will always mean a continuous function. Whenever continuity is not assumed we will use the term function instead.

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Boroński, J.P., Turzański, M. On approximation of asymmetric separators of the n-cube. Fixed Point Theory Appl 2012, 2 (2012). https://doi.org/10.1186/1687-1812-2012-2

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Keywords

  • connected separators
  • algorithm
  • fixed point