Chapter 2 - Convex Functions

Abstract

This chapter is dedicated to convexity. We start by introducing elementary properties of convex sets and functions. In particular, the characterization of convex functions via the convexity of their epigraph. We also study the distance function to a convex set. We then prove that finite dimensional convex functions are continuous. We establish the Line Segment Principle. We also present Minkowski’s Separation Theorem and many of its interesting consequences. Finally, we include in this chapter two results: the properties of the Moreau’s envelope of a convex function, in the finite dimensional case, and the characterization of twice differentiable convex functions.