3 - Convex Functions

Summary

This chapter introduces the concept of a convex function and develops the basic theory of convex functions. Standard continuity and differentiability properties are established. Fundamental notions like subgradients and subdifferentials are introduced and their properties are investigated in detail. Sublinear functions get particular focus, given their recent importance in optimization theory and practice. Some new results on sublinear functions that have never before appeared outside specialized research articles are presented with clean, textbook-style proofs. Elementary Brunn-Minkowski theory is covered, including important consequences like the concavity principle and the Rogers-Shepard inequality.