We start by introducing some standard notation. $\mathbb{R}^{{n+1}}$ is the $(n+1)$-dimensional real space with Euclidean norm and metric. For a subset $A\subset\mathbb{R}^{{n+1}}$ we denote by $\diam(A)$ the diameter of $A$.

The $n$-dimensional simplex $\mathcal{S}_{n}$ is the following subset of $\mathbb{R}^{{n+1}}$

More generally, if $V=\{v_{1},v_{2},\ldots,v_{k}\}$ is a set of vectors, then $S(V)$ is the simplex spanned by $V$:

Let $\mathcal{E}=\{e_{1},e_{1},\ldots,e_{{n+1}}\}$ be the standard orthonormal basis of $\mathbb{R}^{{n+1}}$. So, $\mathcal{S}_{n}$ is the simplex spanned by $\mathcal{E}$. Any element $v$ of $S(V)$ is represented by a vector of coordinates $(\alpha_{1},\alpha_{2},\ldots,\alpha_{k})$ such that $v=\sum_{i}\alpha_{i}v_{i}$; these are called a barycentric coordinates of $v$. If the set $V$ is in general position then the above representation is unique and we say that $V$ is a basis for $S(V)$. If we write $S(V)$ then $V$ is always a basis.

Let $v$ be in $S(V)$, $V=\{v_{1},v_{2},\ldots,v_{k}\}$ a basis and $v$ represented (uniquely) by barycentric coordinates $(\alpha_{1},\alpha_{2},\ldots,\alpha_{k})$. We denote by $F_{V}(v)$ the subset $\left\{j\,|\,\alpha_{j}\neq 0\right\}$ of $\{1,2,\ldots,k\}$ (i.e., the set of non-null coordinates). Let $I\subset\{1,2,\ldots,k\}$, the $I$-th face of $S(V)$ is the set $\{v\in S(V)|F_{V}(v)\subseteq I\}$. A face of $S(V)$ is an $I$-face for some $I$ (note that this is independent of the choice of basis).

The main result we prove is the following:

We prove the KKM Lemma by using Sperner’s Lemma; Sperner’s Lemma is based on the notion of simplicial subdivision and coloring.

An $(n+1)$-coloring of a subdivision $D=(V,\mathcal{B})$ of $\mathcal{S}_{n}$ is a function

A Sperner Coloring of $D$ is an $(n+1)$-coloring $C$ such that $C(v)\in F_{\mathcal{E}}(v)$ for every $v\in V$, that is, if $v$ is on the $I$-th face then its color is from $I$. For example, if $D=(V,\mathcal{B})$ is a subdivision of the standard simplex $\mathcal{S}_{n}$ then the standard basis $\mathcal{E}$ is a subset of $V$ and $F_{\mathcal{E}}(e_{i})=i$. Hence, If $C$ is a Sperner Coloring of $D$ then $C(e_{i})=i$ for all $i=1,2,\ldots,n+1$.

It is a standard result, for example by barycentric subdivisions, that $\mathcal{S}_{n}$ has a sequence of simplicial subdivisions $D_{1},D_{2},\ldots$ such that $|D_{i}|\to 0$. We use this fact to prove the KKM Lemma: