Recommended Citation Gwen Spencer and Francis Edward Su. The LSB theorem implies the KKM lemma. Amer. Math. Monthly, 114(2):156–159, 2007. The LSB Theorem Implies the KKM Lemma (original) (raw)

2016

Abstract

Let Sd be the unit d-sphere, the set of all points of unit Euclidean distance from the origin in Rd+l. Any pair of points in Sd of the form x,-x is a pair of antipodes in Sd. Let Ad be the d-simplex formed by the convex hull of the standard unit vectors in Rd+l. Equivalent^, Ad = {(xu..., xd+x) :? / xt = 1, x {> 0}. The following are two classical results about closed covers of these topological spaces (for the first see [6] or [3], for the second see [5]): The LSB Theorem (Lusternik-Schnirelmann-Borsuk). If Sd is covered by d + 1 closed sets A\\,..., Ad+\\, then some At contains a pair of antipodes. The KKM Lemma (Knaster-Kuratowski-Mazurkiewicz). If Ad is covered by d-+- 1 closed sets C\\, C2,..., Cd+ \\ such that each x in Ad belongs to U{C, : x?> 0}, then the sets C, have a common intersection point (i.e., nd^?C? is nonempty). A cover satisfying the condition in the KKM lemma is sometimes called a KKM cover. It can be described in an alternate way: associate labels 1, 2,..., d...

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