  Whitney's extension problem, i.e., how one can tell whether a function $f : X
\to \mathbb R$, $X \subseteq \mathbb R^n$, is the restriction of a
$C^m$-function on $\mathbb R^n$, was solved in full generality by Charles
Fefferman in 2006. In this paper, we settle the $C^{1,\omega}$-case of a
related conjecture: given that $f$ is semialgebraic and $\omega$ is a
semialgebraic modulus of continuity, if $f$ is the restriction of a
$C^{1,\omega}$-function then it is the restriction of a semialgebraic
$C^{1,\omega}$-function. We work in the more general setting of sets that are
definable in an o-minimial expansion of the real field. An ingenious argument
of Brudnyi and Shvartsman relates the existence of $C^{1,\omega}$-extensions to
the existence of Lipschitz selections of certain affine-set valued maps. We
show that if a definable affine-set valued map has Lipschitz selections then it
also has definable Lipschitz selections. In particular, we obtain a Lipschitz
solution (more generally, $\omega$-H\"older solution, for any definable modulus
of continuity $\omega$) of the definable Brenner-Epstein-Hochster-Koll\'ar
problem. In most of our results we have control over the respective
(semi)norms.
