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

In a recent article (2022) we proved with L. Zaj\'i{c}ek that if G⊂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→ and a concave modulus ω such that limt→∞ω(t)=∞, f is both semiconvex and semiconcave with modulus ω and f∉ C1,ω(G). Here we improve the previous result as follows: If G is as above and ω(t)=tα for some α∈(0,1), then there exists f:G→ that is both semiconvex and semiconcave with modulus ω and f∉ C1,α(G). This result has immediate consequences concerning a first-order quantitative converse Taylor theorem and the problem whether f∈ C1,α(G) whenever f is smooth in a corresponding sense on all lines.