The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

Abstract

We study a variant of the Whitney extension problem (1934) for the space C k,ω (R n ). We identify C k,ω (R n ) with a space of Lipschitz mappings from n into the space P k x R of polynomial fields on n equipped with a certain metric. This identification allows us to reformulate the Whitney problem for C k,ω (R n ) as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of P k x 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 C k,ω (R n ) due to C. Fefferman.