  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.
