Definable Lipschitz selections for affine-set valued maps
摘要
Whitney’s extension problem, i.e., how one can tell whether a function f: X → ℝ, X ⊆ ℝn, is the restriction of a Cm-function on ℝn, was solved in full generality by Charles Fefferman in 2006. In this paper, we settle the C1,ω-case of a related conjecture: given that f is semialgebraic and ω is a semialgebraic modulus of continuity, if f is the restriction of a C1,ω-function then it is the restriction of a semialgebraic C1,ω-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 C1,ω-extensions to the existence of Lipschitz selections of certain affine-set valued maps. We show that if a definable affine-set 4valued map has Lipschitz selections then it also has definable Lipschitz selections. In particular, we obtain a Lipschitz solution (more generally, ω-Hölder solution, for any definable modulus of continuity ω) of the definable Brenner–Epstein–Hochster–Kollár problem. In most of our results we have control over the respective (semi)norms.