Convex sets and convex functions

Abstract

We now give a brief introduction to convex analysis. The chapter is divided into two sections. In Section 2.2, we give some of the most important theorems, namely the separation theorems (sometimes also called Hahn-Banach theorem which is their infinite dimensional version), Carathéodory theorem and Minkowski theorem, also usually better known as Krein-Milman theorem, which is its infinite dimensional version. In Section 2.3, we list some properties of convex functions such as Jensen inequality, the continuity of such functions, the notion of duality and of subdifferential.