Further Abstract Convexity

Abstract

Some definitions related to abstract convexity have been introduced in Chapter 1. In Chapters 2 – 6 we concentrated mainly upon examples of abstract convexity and its applications. In this chapter we continue the examination of abstract convexity in a general situation. For some applications it is convenient to consider abstract convex functions defined only on a subset of the domain of elementary functions. We introduce the notion of abstract convex functions, abstract convex sets and corresponding hulls for this situation and provide many examples of abstract convexity in such a setting. We examine in detail the Fenchel-Moreau conjugacy, subdifferentials and approximate sub differentials (known also as ε-subdifferentials) and present some links between these crucial notions of abstract convexity. We also examine the Minkowski duality, which is a one-to-one correspondence between abstract convex functions and corresponding support sets. This kind of duality has found many applications, some of which are studied in Chapter 8.