A Characterization of Convex Functions

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