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

Let

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 ω

, every function on

G G

which is both semiconvex and semiconcave with modulus

\omega ω

is (globally)

C^{1,\omega} C 1 , ω

-smooth. We show that this result is optimal in the sense that the assumption on

G 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\in C^{1,\omega}(G) f ∈ C 1 , ω ( G )

whenever

f f

is continuous and smooth in a corresponding sense on all lines. We hope that these consequences are of an independent interest.