Convex Sets

Abstract

This chapter summarizes the basic concepts and facts about convex sets. Affine sets, halfspaces, convex sets, convex cones are introduced, together with related concepts of dimension, relative interior and closure of a convex set, gauge and recession cone. Caratheodory’s Theorem and Shapley–Folkman’s Theorem are formulated and proven. The first and second separation theorems are presented and on this basis the geometric structure of a convex set is studied via its supporting hyperplanes, faces, and extreme points. Polars of convex sets and particularly of polyhedral convex sets are introduced and the basic theorem on representation of a polyhedron in terms of its extreme points and extreme directions is established. The chapter closes by a study of systems of convex sets, including a proof of Helly’s Theorem.