
Journal of Convex Analysis 29 (2022), No. 3, 837856 Copyright Heldermann Verlag 2022 Functions on a Convex Set which are both ωSemiconvex and ωSemiconcave Vaclav Krystof Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic krystof@karlin.mff.cuni.cz Ludek Zajícek Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic zajicek@karlin.mff.cuni.cz [Abstractpdf] 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, 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 firstorder 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. Keywords: $\omega$semiconvex functions, $\omega$semiconcave functions, $C^{1, \omega}$smooth functions, smoothness on all lines, converse Taylor theorem, strongly $\alpha(\cdot)$paraconvex functions. MSC: 26B25; 26B35. [ Fulltextpdf (165 KB)] for subscribers only. 