Definable Lipschitz selections for affine-set valued maps

Abstract

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 $C^{1,\ddot{I}‰}$-case of a related conjecture: given that $f$ is semialgebraic and $\ddot{I}‰$ is a semialgebraic modulus of continuity, if $f$ is the restriction of a $C^{1,\ddot{I}‰}$-function then it is the restriction of a semialgebraic $C^{1,\ddot{I}‰}$-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 $C^{1,\ddot{I}‰}$-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, $\ddot{I}‰$-Hölder solution, for any definable modulus of continuity $\ddot{I}‰$) of the definable Brenner-Epstein-Hochster-Kollár problem. In most of our results we have control over the respective (semi)norms.

20 pages; minor changes, Remark 4.10 added; final version