Whitney Extension Theorems for Convex Functions of the Classes C1 and C1,ω. Daniel Azagra and Carlos Mudarra

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