Journal of Convex Analysis 29 (2022), No. 3, 837--856
Copyright Heldermann Verlag 2022
Functions on a Convex Set which are both ω-Semiconvex and ω-Semiconcave
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
[Abstract-pdf]
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 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.
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.
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