A coordinate-free proof of the finiteness principle for Whitney’s extension problem

Abstract

We present a coordinate-free version of Fefferman’s solution of Whitney’s extension problem in the space $C^{m−1,1}(\mathbb R^n)$. While the original argument relies on an elaborate induction on collections of partial derivatives, our proof uses the language of ideals and translation-invariant subspaces in the ring of polynomials. We emphasize the role of compactness in the proof, first in the familiar sense of topological compactness, but also in the sense of finiteness theorems arising in logic and semialgebraic geometry. These techniques may be relevant to the study of Whitney-type extension problems on sub-Riemannian manifolds where global coordinates are generally unavailable.