No abstract

Let C be a subset of R n (not necessarily convex), f : C → R be a function and G : C → R n 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 ∈ C 1,ω (R n ) 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 C 1 convex functions on R n , 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 R n by boundaries of C 1 or C 1,1 convex bodies with prescribed outer normals on K.

Let E be an arbitrary subset of R n (not necessarily bounded)