An elementary direct proof that the Knaster-Kuratowski-Mazurkiewicz lemma implies Sperner's lemma

TL;DR

This paper provides a straightforward, elementary proof demonstrating that the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma implies Sperner's lemma, clarifying a previously less direct logical connection among key economic and topological results.

Contribution

It offers the first elementary direct proof that the KKM lemma implies Sperner's lemma, filling a gap in the literature of foundational mathematical results.

Findings

Established a direct implication from KKM lemma to Sperner's lemma

Simplified the understanding of the relationship among key topological lemmas

Enhanced the toolkit for proofs in economic theory and topology

Abstract

Three central results in economic theory --- Brouwer's fixed-point theorem, Sperner's lemma, and the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma --- are known to be equivalent. In almost all cases, elementary direct proofs of one of these results using any of the others are easily found in the literature. This seems not to be the case for the claim that the KKM lemma implies Sperner's lemma. The goal of this note is to provide such a proof.

Equations

$$S=\mathrm{conv}\{a_1,\dots ,a_{m+1}\}=\left\{{\sum }_{i=1}^{m+1}{\lambda }_i{a}_i\in {\mathbb{R}}^n:{\lambda }_1,\dots ,{\lambda }_{m+1}\ge 0,{\sum }_i{\lambda }_i=1\right\}$$

$$\Delta =\left\{x\in {\mathbb{R}}^n:x_1,\dots ,x_n\ge 0,{\sum }_i{x}_i=1\right\}=\mathrm{conv}\{e_1,\dots ,e_n\}$$

Full text

Keywords. Knaster-Kuratowski-Mazurkiewicz, Sperner, Brouwer.

JEL classification. C62, C69.

To appear in Economics Letters, https://doi.org/10.1016/j.econlet.2017.06.013.

1 Introduction

Three central results for economic theory are known to be equivalent: Brouwer’s (1911) fixed-point theorem, Sperner’s (1928) lemma, and the Knaster-Kuratowski-Mazurkiewicz (KKM, 1929) lemma. Shortly after Sperner’s publication, Knaster et al. (1929, §3–4) showed that Sperner’s lemma implies the KKM lemma and that the latter implies Brouwer’s fixed-point theorem. Much later, Yoseloff (1974, Thm. 1) proved that Brouwer’s fixed-point theorem implies Sperner’s lemma, thereby closing the cycle of implications that makes these results equivalent.

So we have an indirect proof that KKM implies Sperner: KKM implies Brouwer, Brouwer implies Sperner. But my search for an elementary, direct proof came up empty. After recalling relevant definitions, I provide such a proof below. For short direct proofs that Brouwer implies KKM and that Sperner implies Brouwer, see, e.g., Border (1985, p. 44 and 28).

2 Definitions and notation

Throughout the note, sets lie in $\mathbb{R}^{n}$. Denote its standard basis vectors by $e_{1},\ldots,e_{n}$. An $m$-simplex in $\mathbb{R}^{n}$ is the convex hull

[TABLE]

of $m+1$ affinely independent vectors $a_{1},\ldots,a_{m+1}$ in $\mathbb{R}^{n}$, its vertices. Affine independence means that the only scalars for which $\sum_{i=1}^{m+1}\lambda_{i}a_{i}=(0,\ldots,0)$ and $\sum_{i=1}^{m+1}\lambda_{i}=0$ are $\lambda_{1}=\cdots=\lambda_{m+1}=0$. For instance, having $n$ vertices, the unit simplex

[TABLE]

is an $(n-1)$-simplex. Affine independence implies that each element of $S$ has a unique representation as a convex combination $\sum_{i}\lambda_{i}a_{i}$ of its vertices. The scalars $\lambda_{i}$ are called barycentric coordinates or weights. The vertices of $S$ are said to span $S$. A face of a simplex is a simplex spanned by a subset of its vertices. Using the set of all its vertices, $S$ is a face of itself.

A subdivision of $\Delta$ is a finite collection of smaller $(n-1)$-simplices whose union is $\Delta$ and where the intersection of any two such smaller simplices is empty or a face of both. A Sperner labeling assigns a label $1,\ldots,n$ to each vertex of the simplices in the subdivision. The label $\ell(x)$ of each vertex $x$ is chosen among its positive coordinates: $\ell(x)\in\{i:x_{i}>0\}$. A simplex in the subdivision is completely labeled if its set of vertices has all $n$ distinct labels.

The left panel of Figure 1 shows a subdivision of the unit simplex in $\mathbb{R}^{3}$; its corners correspond with the standard basis vectors $e_{1}$, $e_{2}$, and $e_{3}$, and by definition have label 1, 2, and 3, respectively.

3 KKM implies Sperner

Let us start with the formulations of the KKM lemma and Sperner’s lemma:

The KKM lemma**.**

If $C_{1},\ldots,C_{n}$ are closed subsets of $\Delta$ and for each nonempty $J\subseteq\{1,\ldots,n\}$, set $\Delta_{J}:=\mathop{\mathrm{conv}}\{e_{j}:j\in J\}$ is a subset of  $\bigcup_{j\in J}C_{j}$, then  $\bigcap_{i=1}^{n}C_{i}\neq\emptyset$.

Sperner’s lemma**.**

Consider a simplicial subdivision of $\Delta$ and a Sperner labeling. This subdivision contains a completely labeled simplex.

For a direct proof that the KKM lemma implies Sperner’s lemma, we find sets $C_{i}$ in the KKM lemma such that points in their intersection lie in a completely labeled simplex. The intuition is to exploit the unique representation of elements in a simplex as a convex combination of its vertices. If $x$ lies in a simplex and its barycentric coordinate/weight on a vertex $v$ is strictly positive, then that vertex must lie in every face containing $x$: had it been absent, $x$ would have a second representation that did not use $v$. So we take $C_{i}$ to be the elements of simplices in the subdivision whose weights on a vertex with label $i$ are sufficiently large — for convenience, at least $1/n$: since simplices in the subdivision of $\Delta$ have $n$ vertices and weights sum to one, there is always a vertex with weight $1/n$ or more. Now if $x$ belongs to $C_{1}$ and $C_{2}$, it must lie in a face with labels 1 and 2. Repeating this, we find a simplex with all labels. The sets $C_{1}$, $C_{2}$, and $C_{3}$ for our example in $\mathbb{R}^{3}$ are sketched in Fig. 1. To find $C_{1}$, for instance, go through all simplices of the subdivision and color all elements where a vertex with label 1 has barycentric coordinate 1/3 or more. Their union is $C_{1}$.

A more naive approach (yes, my first guess), to define $C_{i}$ as the set of points in simplices with label $i$, does not work. The starred vertex in Fig. 1 lies in a simplex of the subdivision with label $i$, no matter what label $i\in\{1,2,3\}$ you choose, but not in a completely labeled simplex.

Theorem 1**.**

The KKM lemma implies Sperner’s lemma.

Proof.

Define $C_{i}$. Fix label $i\in\{1,\ldots,n\}$. For each simplex $S=\mathop{\mathrm{conv}}\{a_{1},\ldots,a_{n}\}$ in the subdivision, define the possibly empty subset $T_{S}\subseteq S$ of convex combinations $\sum_{j}\lambda_{j}a_{j}$ giving weight $\lambda_{j}\geq 1/n$ to at least one vertex $a_{j}$ with label $i$. Let $C_{i}$ be the union of all these $T_{S}$. As the union of finitely many closed sets, $C_{i}$ is closed.

Verify the KKM condition. Let $J\subseteq\{1,\ldots,n\}$ be nonempty. To show: $\Delta_{J}\subseteq\bigcup_{j\in J}C_{j}$. So let $x\in\Delta_{J}$. This $x$ is a convex combination $x=\sum_{i}\lambda_{i}a_{i}$ of vertices of a simplex $S=\mathop{\mathrm{conv}}\{a_{1},\ldots,a_{n}\}$ of the subdivision. At least one vertex $a_{i}$ has weight $\lambda_{i}\geq 1/n$. Since $x\in\Delta_{J}$, only coordinates of $x$ and hence of $a_{i}$ that lie in $J$ can be positive. By definition of the labeling, $\ell(a_{i})\in J$. So $x\in C_{\ell(a_{i})}\subseteq\bigcup_{j\in J}C_{j}$.

Find a completely labeled simplex. By KKM, there is an $x^{*}\in\bigcap_{i}C_{i}$. By definition of $C_{i}$: for each label $i$, some simplex $S_{i}$ in the subdivision has $x^{*}$ as a convex combination of its vertices with a weight of at least $1/n$ on a vertex $v_{i}$ with label $i$. $S_{1}$ is completely labeled. Its vertex $v_{1}$ has label 1. For labels $p\neq 1$, note that $x^{*}\in S_{1}\cap S_{p}$. This intersection is a face of $S_{1}$ and $S_{p}$. But then $v_{p}$ with label $p$ is one of the vertices (of $S_{1}$ and $S_{p}$) spanning this face: if not, $x^{*}$ would also be a convex combination of vertices of $S_{p}$ other than $v_{p}$, contradicting its unique representation in $S_{p}$. ∎

Acknowledgements

I thank Jörgen Weibull, Henrik Petri, and Albin Erlanson for helpful discussions. Financial support from the Wallander-Hedelius foundation through grant P2010-0094:1 is gratefully acknowledged.