Functions on a convex set which are both $\omega$-semiconvex and $\omega$-semiconcave

TL;DR

This paper proves that functions on certain convex sets that are both semiconvex and semiconcave with the same modulus are necessarily $C^{1, ext{omega}}$-smooth, and shows the optimality of the set assumptions.

Contribution

It establishes the optimal conditions under which functions that are both semiconvex and semiconcave are $C^{1, ext{omega}}$-smooth, with simplified proofs and quantitative versions.

Findings

Functions are $C^{1, ext{omega}}$-smooth under optimal convex set conditions.

Provides direct short proofs and quantitative versions of the main result.

Implications for converse Taylor theorems and line-wise smoothness conditions.

Abstract

Let $G \subset {\mathbb R}^{n}$ be an open convex set which is either bounded or contains a translation of a convex cone with nonempty interior. It is known that then, for every modulus $\omega$, every function on $G$ which is both semiconvex and semiconcave with modulus $\omega$ is (globally) $C^{1,\omega}$-smooth. We show that this result is optimal in the sense that the assumption on $G$ cannot be relaxed. We also present direct short proofs of the above mentioned result and of some its quantitative versions. Our results have immediate consequences concerning (i) a first-order quantitative converse Taylor theorem and (ii) the problem whether $f\in C^{1,\omega}(G)$ whenever $f$ is continuous and smooth in a corresponding sense on all lines. We hope that these consequences are of an independent interest.