A Characterization of Convex Functions

TL;DR

This paper characterizes convex functions on convex subsets of real vector spaces, showing that radial lower semicontinuity combined with a specific midpoint inequality condition is equivalent to convexity.

Contribution

It provides a new characterization of convex functions using radial lower semicontinuity and a midpoint inequality condition.

Findings

Convex functions can be characterized by a radial lower semicontinuity condition.

The midpoint inequality condition is both necessary and sufficient for convexity.

This characterization extends understanding of convex functions in general vector spaces.

Abstract

Let $D$ be a convex subset of a real vector space. It is shown that a radially lower semicontinuous function $f: D\to \mathbf{R}\cup \{+\infty\}$ is convex if and only if for all $x,y \in D$ there exists $\alpha=\alpha(x,y) \in (0,1)$ such that $f(\alpha x+(1-\alpha)y) \le \alpha f(x)+(1-\alpha)f(y)$.