Convex Functions

Abstract

Let X be a convex subset of a vector space V. We say that $f : X\rightarrow \mathbb{R}$ is convex if for all x, y ε X and λ ε (0, 1) we have

$$f(\lambda x + (1 - \lambda )y) \leq \lambda f(x) + (1 - \lambda )f(y).$$

If the inequality is strict when x≠y, then we say that f is strictly convex. In this chapter we aim to look at some properties of these functions, in particular, when E is a normed vector space. For differentiable functions we will obtain a characterization, which will enable us to generalize the concept of a convex function.