When does unique local support ensure convexity?

Abstract:

A basic theorem of convex analysis states that a real-valued function on an open interval of the real line is convex and differentiable if at each point of its domain there exists a unique supporting line. In this paper we show that the same conclusion can be drawn under the weaker hypothesis that there exists a unique locally supporting line at each point. We also show by counterexample that convexity cannot be concluded under analogous circumstances for $f:S \to \mathbb {R}$, where $S \subset {\mathbb {R}^n}$ is open and convex, if $n > 1$.