Whitney's extension problem for multivariate $C^{1,\omega}$-functions

We prove that the trace of the space $C^{1,\omega}({\mathbb R}^n)$to an arbitrary closed subset $X\subset{\mathbb R}^n$is characterized by the following ``finiteness'' property. A function $f:X\rightarrow{\mathbb R}$belongs to the trace space if and only if the restriction $f\vert _Y$ to an arbitrary subset $Y\subset X$ consisting of at most $3\cdot 2^{n-1}$ can be extended to a function $f_Y\in C^{1,\omega}({\mathbb R}^n)$ such that

The constant $3\cdot 2^{n-1}$ is sharp.

The proof is based on a Lipschitz selection result which is interesting in its own right.