A generalized sharp whitney theorem for jets

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