Convex Sets

Abstract

This chapter introduces the basic definitions and facts about convex sets, convex hulls, and affine sets, sometimes also called “flats” in the literature. Three fundamental theorems about convex sets in Rn dealing with the topics of separation, support, and extreme points are proved. Interpreted geometrically, to be convex, a set must contain the line segment connecting any two of its points; to be affine it must contain the whole line through any two of its points. Every affine set is convex, but not conversely. A normed linear space L, along with the trivial examples of the empty set, and sets consisting of one point are both affine and convex. © 1973, Academic Press, Inc.