Smooth selection for infinite sets

Whitney's extension problem asks the following: Given a compact set E?Rn and function f:E?R, how can we tell whether there exists F?Cm(Rn) such that F=f on E? A 2006 theorem of Charles Fefferman [6] answers this question in its full generality. In paper, establish version adapted for variants Whitney problem, including nonnegative extensions smooth selection problems. Among other things, generalize Finiteness Principle by Fefferman-Israel-Luli [9] to setting infinite sets. Our main result is stated terms iterated Glaeser refinement bundle formed taking potential Taylor polynomials at each point E. particular, show bundles (and any with closed, convex fibers) stabilize after bounded number refinements, thus strengthening previous results Glaeser, Bierstone-Milman-Paw?ucki, which only hold affine fibers.