Separation Theorems for Convex Sets

Abstract

Ihe following pages represent an attempt to present a fairly simple exposition of some of the well known basic results on convex sets in real linear spaces having topologies. The results discussed here are those that naturally group themselves around the separation theorem (Theorem 3) and include several of its standard consequences concerning the geometry of convex sets, the Minkowski-Price-6rein-Milman theorem on extreme points, and a version of the so-called fundamental theorem of game theory. Ihe only previous knowledge required here consists essentially of an acquaintance with elementary maneuvers in real linear spaces and in topological spaces; beyond these theHahn43anach theorem and the Hausdorff Maximality Principle are each invokedonce, and some standard criteria for compactness are called upon occasionally. Specifically we shall be concerned with convex and midpoint convex sets in areal linear space having a such that addition and scalar multiplication are continuous in each variable separately.Such spaces have been considered by Nikodym [13] and Klee [8] andhave the advantage that theorems stated in such a context always have as corollaries purely algebraic statements about arbitrary real linear spaces. For in any such space E there is a certain natural topology, called the core topology by Klee [8] and sometimes the radial topology by others, in which a subset A of E is open if and only if for each x in A and each y in E there is some positive real number Ex y such that x + Ay is in A whenever IA| < EX,Y. With respect to this addition and scalar multiplication are continuous in each variable, so that all theorems and corollaries stated below except Corollary 3.13 are true in any real linear space E with respect to this topology. More will be said on this at the end of the paper. The histories of the results stated here are so long and involved that no attempt has been made to describe them. We assert only that almost all of these theorems in their present forms began specifically with Minkowski [12] in his work on convex sets in Euclidean 3-space (Price [14] ), and that this is particularly true of the chief separation theorem (Theorem 3) and the theorem on extreme points (Theorem 4). In view of the present importance of these two results it might, however, be in order to recall briefly that the former was extended to separable Panach Spaces by Ascoli, to normal linear spaces by Mazur,