Whitney extension theorems for convex functions of the classes C1 and C1ω

Abstract

Let C be a subset of ℝn (not necessarily convex), f : C → R be a function and G : C → ℝn be a uniformly continuous function, with modulus of continuity ω. We provide a necessary and sufficient condition on f, G for the existence of a convex function F ∈ CC1ω(ℝn) such that F = f on C and ∇F = G on C, with a good control of the modulus of continuity of ∇F in terms of that of G. On the other hand, assuming that C is compact, we also solve a similar problem for the class of C1 convex functions on ℝn, with a good control of the Lipschitz constants of the extensions (namely, Lip(F) ≲ ∥G∥∞). Finally, we give a geometrical application concerning interpolation of compact subsets K of ℝn by boundaries of C1 or C1,1 convex bodies with prescribed outer normals on K.