Smooth Selection for Infinite Sets

TL;DR

This paper extends Fefferman's 2006 Whitney extension theorem to infinite sets and variants like nonnegative extensions, using Glaeser refinement to analyze the stabilization of bundles with convex fibers.

Contribution

It generalizes the Finiteness Principle for smooth selection to infinite sets and introduces a stabilization result for bundles with convex fibers.

Findings

Bundles with convex fibers stabilize after finite Glaeser refinements

Generalization of the Finiteness Principle to infinite sets

Strengthens previous results by Glaeser, Bierstone-Milman-Pawłucki, and Fefferman

Abstract

Whitney's extension problem asks the following: Given a compact set $E\subset\mathbb{R}^n$ and a function $f:E\to \mathbb{R}$, how can we tell whether there exists $F\in C^m(\mathbb{R}^n)$ such that $F=f$ on $E$? A 2006 theorem of Charles Fefferman \cite{F06} answers this question in its full generality. In this paper, we establish a version of this theorem adapted for variants of the Whitney extension problem, including nonnegative extensions and the smooth selection problems. Among other things, we generalize the Finiteness Principle for smooth selection by Fefferman-Israel-Luli \cite{FIL16} to the setting of infinite sets. Our main result is stated in terms of the iterated Glaeser refinement of a bundle formed by taking potential Taylor polynomials at each point of $E$. In particular, we show that such bundles (and any bundles with closed, convex fibers) stabilize after a bounded number of Glaeser refinements, thus strengthening the previous results of Glaeser, Bierstone-Milman-Paw{\l}ucki, and Fefferman which only hold for bundles with affine fibers.