Functions on a convex set which are both $ ω$-semiconvex and $ ω$-semiconcave II

Abstract

In a recent article (2022) we proved with L. Zajíček that if $ G\subset\R^n $ is an unbounded open convex set that does not contain a translation of a convex cone with non-empty interior, then there exist $ f:G\to\R $ and a concave modulus $ ω$ such that $ \lim_{t\to\infty}ω(t)=\infty $, $ f $ is both semiconvex and semiconcave with modulus $ ω$ and $ f\notin C^{1,ω}(G) $. Here we improve the previous result as follows: If $ G $ is as above and $ ω(t)=t^α $ for some $ α\in(0,1) $, then there exists $ f:G\to\R $ that is both semiconvex and semiconcave with modulus $ ω$ and $ f\notin C^{1,α}(G) $. This result has immediate consequences concerning a first-order quantitative converse Taylor theorem and the problem whether $ f\in C^{1,α}(G) $ whenever $ f $ is smooth in a corresponding sense on all lines.

Theorems & Definitions (45)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Lemma 2.5
• proof
• Remark 2.6
• Remark 2.7