Convexity in Linear Spaces

Abstract

Our purpose in this first chapter is to establish the basic terminology and properties of convex sets and functions, and of the associated geometry. All concepts are “primitive”, in the sense that no topological notions are involved beyond the natural (Euclidean) topology of the scalar field. The latter will always be either the real number field R, or the complex number field C. The most important result is the “basic separation theorem”, which asserts that under certain conditions two disjoint convex sets lie on opposite sides of a hyperplane. Such a result, providing both an analytic and a geometric description of a common underlying phenomenon, is absolutely indispensible for the further development of the subject. It depends implicitly on the axiom of choice which is invoked in the form of Zorn’s lemma to prove the key lemma of Stone. Several other equally fundamental results (the “support theorem”, the “subdifferentiability theorem”, and two extension theorems) are established as equivalent formulations of the basic separation theorem. After indicating a few applications of these ideas we conclude the chapter with an introduction to the important notion of extremal sets (in particular extreme points) of convex sets.