Ja n 20 06 The Whitney extension problem and Lipschitz selections of set - valued mappings in jet - spaces

We study a variant of the Whitney extension problem [21, 22] for the space C k,ω (R n). We identify C k,ω (R n) with a space of Lipschitz mappings from R n into the space P k × R n of polynomial fields on R n equipped with a certain metric. This identification allows us to reformulate the Whitney problem for C k,ω (R n) as a Lip-schitz selection problem for set-valued mappings into a certain family of subsets of P k × R n. 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.