Charles Explorer logo
🇨🇿

Functions on a Convex Set which are both omega-Semiconvex and omega-Semiconcave

Publikace na Matematicko-fyzikální fakulta |
2022

Tento text není v aktuálním jazyce dostupný. Zobrazuje se verze "en".Abstrakt

Let G subset of 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,C-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 is an element of C-1,C-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.