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

Václav Kryštof; Luděk Zajíček

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arXiv · 2103.00524

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

Václav Kryštof·Luděk Zajíček

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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.

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BibTeX RIS

Václav Kryštof, Luděk Zajíček. 2021-02-28. Functions on a convex set which are both $\omega$-semiconvex and $\omega$-semiconcave. https://arxiv.org/abs/2103.00524

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