A Characterization of Convex Functions

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 $f(\alpha x+(1-\alpha )y)\le \alpha f(x)+(1-\alpha )f(y)$.