Classically, a twice differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ is convex if and only if the second derivative is nonnegative everywhere. In this recent question, Derivative of Convex Functional, it's shown that the same result holds for twice Frechet differentiable functionals on Banach spaces, $f:X\rightarrow \mathbb{R}$.

In both these cases, we have a result saying something to the effect of: "local convexity implies global convexity". My question is, how far can this idea be generalized?

Hypothesis: Let $C$ be a connected subset of a topological vector space, and let $\{ U_\alpha \}_{\alpha \in A}$ be an open cover of the boundary $\partial C$. If $U_\alpha \cap C$ is convex for all $\alpha \in A$, then $C$ is convex.

Informally, "Inspect the boundary of a connected set with a (variable-size) magnifying glass. If, everywhere you look, it looks convex, then the set is globally convex." Note the connectedness condition has to come into play somewhere - consider two disjoint sphere sitting in space.

A special case I've been considering is where the space is Banach, and the set's boundary is path connected and compact. In this case I think it's true but the proof is elusive.. In the general case I'm not so sure.