The Whitney Extension Problem and Lipschitz Selections of Set-valued Mappings in Jet-spaces

We study a variant of the Whitney extension problem (1934) for the space Ck,ω(Rn). We identify Ck,ω(Rn) with a space of Lipschitzmappings from Rn into the space Pk × Rn of polynomial fields on Rn equipped with a certain metric. This identification allows us to reformulate the Whitney problem for Ck,ω(Rn) as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of Pk×R. We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for Ck,ω(Rn) due to C. Fefferman. 1. The main problem and main results Let ω : R+ → R+ be a continuous concave function satisfying ω(0) = 0. We let Ċ(R) denote the space of all functions f : R → R with continuous derivatives of all orders up to k, for which the seminorm