A generalized sharp Whitney theorem for jets

Suppose that, for each point x in a given subset E subset of R-n, we are given an m-jet f (x) and a convex, symmetric set sigma(x) of m-jets at x. We ask whether there exist a function F epsilon C-m,C-w(R-n) and a finite constant M, such that the m-jet of F at x belongs to f (x) + M sigma(x) for all x epsilon E. We give a necessary and sufficient condition for the existence of such F, M, provided each sigma(x) satisfies a condition that we call "Whitney w-convexity".