  Let $G \subset {\mathbb R}^{n}$ be an open convex set which is either bounded
or contains a translation of a convex cone with nonempty interior. It is known
that then, for every modulus $\omega$, every function on $G$ which is both
semiconvex and semiconcave with modulus $\omega$ is (globally)
$C^{1,\omega}$-smooth. We show that this result is optimal in the sense that
the assumption on $G$ 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 $f\in C^{1,\omega}(G)$ whenever $f$
is continuous and smooth in a corresponding sense on all lines. We hope that
these consequences are of an independent interest.
