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  • Sperners Lemma Implies Kakutanis FixedPoint Theorem

    Mutiara Sondjaja

    Francis E. Su, Advisor

    Jon Jacobsen, Reader

    May, 2008

    Department of Mathematics

  • Copyright c 2008 Mutiara Sondjaja.

    The author grants Harvey Mudd College the nonexclusive right to make this workavailable for noncommercial, educational purposes, provided that this copyrightstatement appears on the reproduced materials and notice is given that the copy-ing is by permission of the author. To disseminate otherwise or to republish re-quires written permission from the author.

  • Abstract

    Kakutanis fixed point theorem has many applications in economics andgame theory. One of its most well-known applications is in John Nashspaper [8], where the theorem is used to prove the existence of an equilib-rium strategy in n-person games. Sperners lemma, on the other hand, isa combinatorial result concerning the labelling of the vertices of simplicesand their triangulations. It is known that Sperners lemma is equivalent toa result called Brouwers fixed point theorem, of which Kakutanis theoremis a generalization.

    A natural question that arises is whether we can prove Kakutanis fixedpoint theorem directly using Sperners lemma without going through Brou-wers theorem. The objective of this thesis to understand Kakutanis the-orem, Sperners lemma, and how they are related. In particular, I exploreways in which Sperners lemma can be used to prove Kakutanis theoremand related results.

  • Contents

    Abstract iii

    Acknowledgments ix

    1 Introduction 1

    2 Background 52.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Kakutanis Fixed Point Theorem . . . . . . . . . . . . . . . . 72.3 Sperners Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 20

    3 Sperners Lemma and Kakutanis Fixed Point Theorem 273.1 Sperners Lemma Implies Weak Kakutanis Theorem . . . . 273.2 Preliminary Approaches . . . . . . . . . . . . . . . . . . . . . 333.3 Sperners Lemma Implies Kakutanis Fixed Point Theorem . 373.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    4 Polytopal Sperner Lemma and von Neumann Intersection Lemma 434.1 The Case in RR . . . . . . . . . . . . . . . . . . . . . . . . 444.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    5 Discussion and Future Work 495.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

    A Proof of Approximation Lemma 51

    Bibliography 53

  • List of Figures

    1.1 An illustration of Brouwers fixed point theorem . . . . . . . 21.2 An illustration of Sperners Lemma in 2 dimensions . . . . . 3

    2.1 An illustration of Cohens proof . . . . . . . . . . . . . . . . . 222.2 An illustration of Cohens labelling . . . . . . . . . . . . . . . 232.3 An illustration of Polytopal Sperners Lemma . . . . . . . . 242.4 An illustration of Proposition 2.12 . . . . . . . . . . . . . . . 25

    3.1 An illustration of the labelling method . . . . . . . . . . . . . 293.2 An illustration of labelling using Method 2 . . . . . . . . . . 353.3 An illustration of Example 3.6 . . . . . . . . . . . . . . . . . . 363.4 An illustration of Example 3.7 . . . . . . . . . . . . . . . . . . 373.5 An illustration of Lemma 3.8 . . . . . . . . . . . . . . . . . . 38

    4.1 An illustration of Lemma 4.1 in RR . . . . . . . . . . . . . 44

  • Acknowledgments

    I would like to thank Professor Su for his utmost guidance, patience, andencouragement in all aspects of this project, and for being my mentor ingeneral. I would like to thank Professor Jacobsen for his encouragementand very helpful feedback on drafts of this thesis.

    I would also like to thank everyone in the math department, for teach-ing me mathematics and for all the support they have provided me in thepast four years.

    To my parents, my sister, and my friends, thank you for being there.

  • Chapter 1

    Introduction

    In 1912, L. E. J. Brouwer proved a fixed point theorem for continuous func-tions from a compact, convex set in Rn to itself. Brouwers fixed pointtheorem states that given a closed convex set S Rn and a continuousfunction f from S to itself, there exists a point x in S such that f (x) = x. Wecall such point x a fixed point of f .

    To give a simple illustration of Brouwers fixed point theorem, considerthe case when S is the closed interval [0, 1] R. Then, it is intuitively veryeasy to see that for any continuous function f from [0, 1] to itself, the graphof y = f (x) must either cross or touch the line x = y at least once (seeFigure 1.1).

    There are many generalizations of Brouwers fixed point theorem, in-cluding a fixed point theorem proved by Shizuo Kakutani in 1941 [7]. Kaku-tanis fixed point theorem considers upper semicontinuous, compact andconvex-valued point-to-set mappings instead of continuous functions as inBrouwers theorem. We define the terminology and discuss the concepts ingreater detail in the next chapter.

    In this thesis, I explore the relationship between Kakutanis theoremand a result from combinatorics called Sperners lemma, which is a state-ment about the labelling of the vertices of a d-simplex (a d-dimensional tri-angle) and its subdivision into smaller simplices. (We call such subdivisiona triangulation.)

    In two dimensions, Sperners lemma can be stated as follows: Considera triangle (a 2-simplex) S whose three vertices are labelled 0, 1, and 2. Trian-gulate S into smaller triangles, such that any two small triangles are eitherdisjoint, share a vertex, or share an edge. Label the new vertices resultingfrom this triangulation in any way such that if a vertex lies on the edge S

  • 2 Introduction

    Figure 1.1: Illustration of Brouwers fixed point theorem on [0, 1]

    with endpoints labelled i and j (where i, j {0, 1, 2} and are distinct), thenthis vertex may be labelled i or j only. If a vertex lies in the interior of S, itmay receive any label. Then, Sperners lemma states that there is an oddnumber of small triangles in the triangulation of S whose vertices receivethree distinct labels 0, 1, and 2. We call such triangles completely-labelled.In particular, there is always at least one completely-labelled triangle. Weillustrate this in Figure 1.2.

    Sperners lemma is equivalent to Brouwers fixed point theorem as wellas to a number of other related results, such as the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma [6]. In this case, by equivalent, we mean thatone result implies the other and is implied by the other without using ad-ditional results other than classical results of analysis such as convergenceof a subsequence of any sequence of points in compact spaces.

    The equivalence among the results mentioned above motivates the fol-lowing question. Since Kakutanis fixed point theorem is a generalizationof Brouwers fixed point theorem, which is equivalent to Sperners lemma,could we prove Kakutanis theorem directly from Sperners lemma? Infact, this question also has practical significance. Algorithms for findingcompletely-labelled simplices, whose existence is guaranteed by Spernerslemma, are very well developed. As we will see later, the proof of Brou-wers theorem from Sperners lemma is constructive. Hence, a constructiveproof of Kakutanis theorem directly from Sperners lemma could provideinsights into building efficient algorithms for finding fixed points of thepoint-to-set mappings specified in the hypothesis of Kakutanis theorem.

  • 3

    Figure 1.2: An illustration of Sperners Lemma in 2 dimensions

    In a paper published in 1980, D. I. A. Cohen presented a proof of Kaku-tanis theorem directly from Sperners lemma without going through Brou-wers fixed point theorem [4], but this proof seems to be incomplete. Ex-actly how this proof is incomplete will be discussed in Section 2.4.1. Themain result of this thesis is in Chapter 3, where we present a proof of Kaku-tanis fixed point theorem directly from Sperners lemma. In this chap-ter, we also present preliminary approaches that we have considered forcompleting Cohens proof. In Chapter 4, we also explored approaches forproving a related result called von Neumanns intersection lemma, whichis known to be equivalent to Kakutanis theorem, using a polytopal gener-alization of Sperners lemma in [6].

  • Chapter 2

    Background

    This chapter provides the necessary background information on Kakutanisfixed point theorem and Sperners lemma. We first establish some basic ter-minology in Section 2.1. Section 2.2 provides background information onKakutanis fixed point theorem, including the original proof using Brou-wers fixed point theorem. In Section 2.3, we state Sperners lemma andshow the equivalence between the lemma and Brouwers fixed point theo-rem.

    2.1 Terminology

    In this section, we define some basic terminology that we will frequentlyuse in the paper:

    Graph For any function : X Y, the graph of , denoted Gr (), is justgiven by our usual notion of a graph of a function. We formally stateit as

    Gr () = {(x, y) XY|y = f (x)}.

    Similarly, for any point-to-set mapping f : X 2Y, we define thegraph of f , denoted Gr ( f ) as

    Gr ( f ) = {(x, y) XY|y f (x)}.

    Convex We call a set S convex if given any two points x, y S, the linesegment between them is completely contained in S.

  • 6 Background

    Convex combination Given a set of points v1, . . . , vd, and scalars 1, . . . , d,such that

    d

    i=1

    i = 1

    andi 0,

    then the pointd

    i=1

    ivi is called a convex combination o

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