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

We study a variant of the Whitney extension problem for the space $C^{k,\omega}(R^n)$. We identify this space with a space of Lipschitz mappings from $R^n$ into the space $P_k \times R^n$ of polynomial fields on $R^n$ equipped with a certain metric. This identification allows us to reformulate the Whitney problem for $C^{k,\omega}(R^n)$ as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of $P_k \times R^n$. 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 $C^{k,\omega}(R^n)$ due to C. Fefferman.