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

Abstract

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