The LSB Theorem Implies the KKM Lemma
Gwen Spencer, Francis Edward Su · American Mathematical Monthly · 2007
Let S d be the unit d-sphere, the set of all points of unit Euclidean distance from the origin in R d+1. Any pair of points in S d of the form x, −x is a pair of antipodes in S d. Let ∆ d be the d-simplex formed by the convex hull of the standard unit vectors in R d+1. Equivalently, ∆ d = {(x1,..., xd+1): i xi = 1, xi ≥ 0}. The following are two classical results about closed covers of these topological spaces: The LSB Theorem (Lusternik-Schnirelman-Borsuk [6, 3]). Suppose that S d is covered by d + 1 closed sets A1,..., Ad+1. Then some Ai contains a pair of antipodes. The KKM Lemma (Knaster-Kuratowski-Mazurkiewicz [5]). Suppose that ∆ d is covered by d + 1 closed sets C1, C2,...Cd+1 such that for each x in ∆ d, x is in ∪{Ci: xi> 0}. Then all the sets have a common intersection point, i.e., ∩ d+1 i=1 Ci is non-empty. A cover satisfying the condition in the KKM lemma is sometimes called a KKM cover. It can be rephrased in an alternate way: associate labels 1, 2,.., d+1 to the vertices of ∆ d; then demand that vertex i is covered by set Ci and that each face of ∆ d is covered by the sets that correspond to the vertices spanning that face.