Functions on a Convex Set which are Both ω-Semiconvex and ω-Semiconcave II

In a recent article [Functions on a convex set which are both

-semiconvex and

\omega ω

-semiconcave I, J. Convex Analysis 29 (2022) 837–856] we proved with L. Zajíček that if

G\subset\mathbb{R}^n G ⊂ 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} f : G → R

and a concave modulus

\omega ω

such that

\lim_{t\to\infty}\omega(t)=\infty lim ⁡ t → ∞ ω ( t ) = ∞

,

f f

is both semiconvex and semiconcave with modulus

\omega ω

and

f\notin C^{1,\omega}(G) f ∉ C 1 , ω ( G )

. Here we improve the previous result as follows: If

G G

is as above and

\omega(t)=t^{\alpha} ω ( t ) = t α

for some

\alpha\in(0,1) α ∈ ( 0 , 1 )

, then there exists

f:G\to\mathbb{R} f : G → R

that is both semiconvex and semiconcave with modulus

\omega ω

and

f\notin C^{1,\alpha}(G) f ∉ 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,\alpha}(G) f ∈ C 1 , α ( G )

whenever

f f

is smooth in a corresponding sense on all lines.