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