A Proof of Atanassov's Conjecture and Other Generalizations of Sperner's Lemma
aa r X i v : . [ m a t h . C O ] M a y A PROOF OF ATANASSOV’S CONJECTURE AND OTHERGENERALIZATIONS OF SPERNER’S LEMMA.
YITZCHAK SHMALO
Abstract.
A simple proof of Atanassov’s Conjecture is presented. Atanassov’sConjecture is a generalization of Sperner’s Lemma, a lemma which has been usedto prove Brouwer’s Fixed Point Theorem, among other fixed point theorems. Theproof of Atanassov’s Conjecture is based on the Brouwer Degree of maps and isextremely elementary. It is much simpler than the original proofs given for theconjecture and provides some insight into the nature of the conjecture. Furthermore,a generalization of the conjecture is presented and finally a new theorem, similar tothe original Sperner Lemma, is proved. Introduction
In the beginning of the twentieth century, Luitzen Brouwer proved the Brouwer FixedPoint Theorem, which states that every continuous function f from an n -dimensionalball to itself has a fixed point, meaning that for some x we have f p x q “ x . Historically,Sperner’s Lemma has been used to provide a straightforward proof for Brouwer’s FixedPoint Theorem. Some have also used Sperner’s Lemma to prove Kakutani Fixed PointTheorem, a generalization of Brouwer’s Fixed Point Theorem. Such proofs are generallysimpler than those previously offered for the existence of fixed points and are alsosomewhat constructive, meaning that they can be used to approximate the positionof fixed points. Sperner’s Lemma itself is also simple to prove and much is knownabout generalizations of the lemma, for example Atanassov’s Conjecture. In this paper,proofs for Atanassov’s Conjecture and other propositions similar to Sperner’s Lemmaare given.1.1. Sperner’s Lemma for simplices.
Consider an n -simplex T , and T “ Ť i P I T i ,with I finite, a simplicial decomposition of T . A labeling of T is a map φ : T Ñt , , . . . , n ` u . In particular, each vertex v of T and of any of the T i ’s is assigned aunique label φ p v q P t , , . . . , n ` u .A simplex T is said to be completely labeled if its vertices are assigned all labels from t , , . . . , n ` u .The labeling φ is called a Sperner labeling if every point p that lies on some face S of T is assigned one of the labels of the vertices of S . There are no restrictions on thelabeling of the interior points of T .In the most basic form the Sperner Lemma states the following: Theorem 1.1 (Sperner Lemma [Spe28]) . Given an n -simplex T with a completeSperner labeling, a simplicial decomposition T “ Ť i T i . Then the number of com-pletely labeled simplices T i is odd. In particular, there exists at least one simplex T i that is completely labeled. Atanassov’s Conjecture is a generalization of the lemma which states the following.A simplex P i is completely labeled if it has vertices carrying d ` P be a convex, d -dimensional polytope, with n vertices, which is divided into convex, d -dimensional simplices t P i u i P I , with I finite, such that P “ Ť i P I P i , and for i ‰ j ,int p P i q X int p P j q “ H and P i X P j is either empty or a face of both P i and P j . Assumethat φ : V p P q Y Ť i P I V p P i q Ñ t , , . . . , n u is a labeling of the vertices of P and of P i such that: ‚ Each label from t , , . . . , n u is assigned to some vertex of P , so no two verticesare assigned the same label; ‚ If a vertex of P i lies on some face W of P , then that vertex is assigned one ofthe labels of the vertices of W .Then there are at least n ´ d simplices of t P i u i P I ; each completely labeled. Thistheorem was first proved in [LPS02].2. Preliminaries
Sperner’s Lemma for polytopes.
First we describe a more general SpernerLemma-type of result for polytopes, following [BN98]. Parts of this section were takenalmost directly from [GS17]. Let P be a convex n -dimensional polytope. We considera labeling φ : P Ñ t , , . . . , n ` u of P .A polytope is said to be completely labeled if its vertices are assigned all labels from t , , . . . , n ` u .A labeling φ : P Ñ t , . . . , n ` u is said to be a non-degenerate labeling if no p n ´ q -dimensional face of P contains points which take p n ` q or more different values. Thus,every n -dimensional face of P can only contains points which take p n ` q or less differentlabels.We introduce some tools.Consider the standard n -dimensional simplex T “ conv p , e , e , . . . , e n q Ă R n ,where we denote by p e , e , . . . , e n q the standard basis of R n , and by conv p¨q the convexhull of a set. We denote the corresponding vertices of T as t a , . . . , a n ` u .Given an n -dimensional, oriented, convex polytope P , a labeling φ : P Ñ t , . . . , n ` u , and the standard n -simplex T of vertices a , . . . , a n ` , a realization of φ is a con-tinuous map f : P Ñ T , satisfying the following conditions:(i) If v is a vertex of P then f p v q “ a φ p v q , i.e., f p v q is the vertex a i of T with theindex i equal to the label of v ;(ii) If S is face of P with vertices v , . . . , v k , then f p S q Ă conv p a φ p v q , . . . , a φ p v k q q .Informally, a realization of P is a continuous mapping of P onto T that ‘wraps’ B P around B T , such that the labels of the vertices of P match with the indices i of thevertices of T . Such f is in general non-injective. For a smooth, boundary preservingmap f : p M, B M q Ñ p N, B N q between two oriented n -dimensional manifolds with IMPLE PROOF OF ANASSOV’S CONJECTURE 3 boundary, it is known that deg p f q “ deg pB f q , where B f : B M Ñ B N is the map inducedby f on the boundaries. The degree of the map f can be defined as the signed numberof preimages f ´ p p q “ t q , . . . , q k u of a regular value p of the map f , where each point q i is counted with a sign ˘ df q i : T q i M Ñ T p N is orientationpreserving or orientation reversing. That is, deg p f q “ ř q P f ´ p p q sign p det p df q qq , where p P N zB N is a regular value of f . The definition of the Brouwer degree extends viahomotopy to continuous maps. Proposition 2.1 ([BN98]) . Let P be a convex n -dimensional polytope. (i) Any labeling φ of some P admits a realization f ; (ii) Any two realizations of the same labeling are homotopic as maps of pairs p P, B P q ÞÑ p T, B T q ; (iv) If deg p f q ‰ then P is completely labeled, where deg p f q is the Brouwer Degreeof the realization of φ , a labeling of P . The following is a generalization of the Sperner Lemma from [BN98].
Theorem 2.2 ([BN98]) . Assume that P is an n -dimensional polytope, P “ Ť i P I P i isa decomposition of P into polytopes as above, and φ : P Ñ t , . . . , n ` u is a non-degenerate labeling . If deg pB f q ‰ , then there exists a polytope P i that is completelylabeled.Proof. Let f : P Ñ T be a realization of the labeling φ ; the existence of f is ensured byProposition 2.1. We transform the polytope P into another polytope P ˚ , homotopicallytransform the decomposition P “ Ť i P I P i into another decomposition P ˚ “ Ť i P I P ˚ i ,and homotopically transform the map f : P Ñ T into a map f ˚ : P ˚ Ñ T as follows: ‚ We construct a new polytope P ˚ by appending to the vertices of P all thevertices of the the P i ’s lying on the faces of P , and appending to the faces of P all the faces of the P i ’s lying on the faces of P ; the resulting polytope P ˚ stillhas the decomposition P ˚ “ Ť i P I P i ; ‚ We apply a homotopy deformation to the decomposition Ť i P I P i and to therealization f to obtain a new decomposition P ˚ “ Ť i P I P ˚ i , with the labelingof P ˚ i inherited from that of P i , and a new realization f ˚ : P ˚ Ñ T so that, forevery i , the restriction of f ˚| P ˚ i to P ˚ i maps B P ˚ i to B T and is also a realization.The later property ensures that deg p f ˚| P ˚ i q “ deg pB f ˚|B P ˚ i q .From the condition that if a vertex of P ˚ i lies on some face S of P ˚ , then that vertexis assigned one of the labels of the vertices of S , we have deg p f ˚ q “ deg p f q . Using theaddition and homotopy properties of the Brouwer degree, we obtain thatdeg pB f q “ deg p f q “ deg p f ˚ q “ ÿ i deg p f ˚ i q “ ÿ i deg pB f ˚ i q If deg p f q ‰
0, then ř i deg pB f ˚ i q ‰
0, which means that there exists a polytope P ˚ i such that deg pB f ˚|B P ˚ i q ‰
0, hence deg p f ˚| P ˚ i q ‰
0. By Proposition 2.1, and sincethe labeling of P i is the same as the labeling of P ˚ i , we have that P i is completelylabeled. (cid:3) YITZCHAK SHMALO
Now we give the notion of a cover of a polytope, following [LPS02]. A cover C ofa convex polytope P is a collection of simplices in P such that Ť S P C S “ P . We saythat a labeled set of simplices cover a polytope P , or is a cover of P , under a map f ,if the following conditions hold:(i) If v is a vertex of S in the collection then f p v q “ a φ p v q , i.e., f p v q is the vertex a i of P with the index i equal to the label of v ;(ii) If R is face of S with vertices v , . . . , v k , then f p R q Ă conv p a φ p v q , . . . , a φ p v k q q .(ii) f is surjective. Theorem 2.3 ([LPS02] and [RS85]) . Let C p P q denote the covering number of an p n, p q -polytope P ; which is the size of the smallest cover of P . Then, C p P q ě p n ´ d q .This result is best possible as the equality is attained for stacked polytopes. Proof of Atanassov’s Conjecture
Theorem 3.1.
Let P be a convex, d -dimensional polytope, with n vertices, whichis divided into convex, d -dimensional simplices t P i u i P I , with I finite, such that P “ Ť i P I P i , and for i ‰ j , int p P i q X int p P j q “ H and P i X P j is either empty or a face ofboth P i and P j . Assume that φ : V p P q Y Ť i P I V p P i q Ñ t , , . . . , n u is a labeling of thevertices of P and of P i such that: ‚ Each label from t , , . . . , n u is assigned to some vertex of P , so no two verticesare assigned the same label; ‚ If a vertex of P i lies on some face W of P , then that vertex is assigned one ofthe labels of the vertices of W .A simplex P i is completely labeled if it has vertices carrying d ` different labels.Atanassov’s Conjecture states that there are at least n ´ d simplices P i ; each completelylabeled.Proof. Denote by a i the vertex of P labeled with i . Form a realization of φ as acontinuous map f : P Ñ P , satisfying the following conditions: ‚ If v is a vertex of P then f p v q “ a φ p v q , i.e., f p v q is the vertex of P of indexequal to the label of v ; ‚ If R is face of P with vertices w , . . . , w k then f p R q Ă conv p a φ p w q , . . . , a φ p w k q q ,where conv p¨q denotes the convex hull of a set. ‚ For any i , the restriction of f | i to P i , with P i having vertices l , . . . , l k , is suchthat f p P i q Ă conv p a φ p l q , . . . , a φ p l k q q and is also a realization.Given that a vertex of P i which lies on some face W of P is assigned one of the labelsof the vertices of W , we have that this labeling is non-degenerate . From the conditionsof the realization and the first condition of the labeling, we have that deg pB f q “ deg p f q “
1. Thus, f is surjective, which means that a set of P i simplices cover P under f . Notice, however, that every P i which is not completely labeled is mapped to a n ´ P , under the realization, and cannot be used to cover P . Soonly completely labeled P i polytopes can be used to cover P . But the smallest numberof simplices needed to cover a d -dimensional polytope with n vertices is n ´ d . Thuswe must have at least n ´ d complete simplices P i . P must be split into at least n ´ d IMPLE PROOF OF ANASSOV’S CONJECTURE 5 different simplices and each one must be mapped to by at least one P i and each such P i is completely labeled. (cid:3) Remark . After writing this paper I found a similar proof in [Mus14], one whichalso uses the notion of a degree. Nevertheless, the use of the equality deg pB f q “ deg p f q greatly simplifies the proof.The following is a generalizations of Atanassov’s conjecture. Theorem 3.3.
Let P be a convex, d -dimensional polytope, with n vertices, P “ Ť i P I P i is a decomposition of P into simplices as above. Assume that φ : V p P q Y Ť i P I V p P i q Ñt , , . . . , n u is a non-degenerate labeling of the vertices of P and of P i . A simplex P i is completely labeled if it has vertices carrying d ` different labels. Take P to be a d -dimensional polytope, with n vertices. We require that each label from t , , . . . , n u is assigned to some vertex of P , so no two vertices are assigned the same label. Take f to be a realization of φ , where f : P Ñ P . Then we have at least pp n ´ d qqp deg pB f qq different completely labeled simplices P i . Furthermore, if deg pB f q “ then we will havean even number of completely labeled simplices.Proof. Follows almost immediately from the previous proof. (cid:3) Neighboring Labeling Lemma
In this section a new labeling lemma is presented, one very similar to the originalSperner Lemma. The concept of the Brouwer Degree is again used.
Theorem 4.1.
Let P be a convex, d -dimensional polytope, with n vertices, whichis divided into convex, d -dimensional polytopes t P i u i P I , with I finite, such that P “ Ť i P I P i , and for i ‰ j , int p P i q X int p P j q “ H and P i X P j is either empty or a face ofboth P i and P j . Assume that φ : V p P q Y Ť i P I V p P i q Ñ t , , . . . , n u is a labeling of thevertices of P and of P i such that: ‚ Each label from t , , . . . , n u is assigned to some vertex of P , so no two ver-tices are assigned the same label. We call two labels d -similar if there is a d dimensional face of P which they share. Also, a label is n -similar to itself forall n . ‚ Every vertex v of every P i is assigned one of the labels of the vertices of P inthe car p v q . ‚ If a vertex of P i shares a k -face with another vertex (either of P i or of P j ) thenboth of their labels must be k -similar, for all k .A polytope P i is completely labeled if it has vertices carrying labels t , , . . . , n u . Thislemma states that there is at least one completely labeled P i .Proof. We form a realization f : P Ñ P of φ . Any P i is either mapped only to B P orcovers P entirely. This observation follows from the condition that if a vertex of P i isconnected to another vertex (either of P i or of P j ) then both of their labels must besimilar. Once again we have that deg pB f q “ deg p f q “
1. But if deg p f q “ P i , for any individual P i either covers P entirely, in which YITZCHAK SHMALO case it is completely labeled, or only covers the boundary of P . But if all of the P i ’sonly cover the boundary of P then deg p f q “ (cid:3) Acknowledgement
The author is grateful to Marian Gidea, who read and commented on the first draftof this work. As mentioned, parts of the second section were taken from a paper whichwe both authored.
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