Structure of Convex Sets and Functions

Abstract

This chapter is devoted to investigating the deeper properties of convex sets and convex functions on fairly general affine and vector spaces. The separation properties of convex sets are very important in optimization, especially in duality theory, with more sophisticated separation theorems leading to better duality results; see for example Theorem 11.15, on the strong duality in convex programming. In turn, sophisticated separation theorems are obtained by a careful study of the properties of interior points of convex sets. Since a convex set is not necessarily full-dimensional, it is important to study the “relative interior” of convex sets. It turns out that the relative interior of a convex set can be studied in a purely algebraic setting, without any use of topological notions, and this leads to a rich theory of the relative algebraic interior of convex sets. If the space under consideration has a topology, then it turns out that there is a strong connection between the algebraic and topological notions of relative interior, and the topological results can be obtained from the corresponding algebraic ones with relative ease.