An Extension Theorem for convex functions of class C1,1 on Hilbert spaces

Let H be a Hilbert space, E subset of H be an arbitrary subset and f : E -> R, G : E -> H be two functions. We give a necessary and sufficient condition on the pair (f, G) for the existence of a convex function F is an element of C-1,C-1 (H) such that F = f and del F = G on E. We also show that, if this condition is met, F can be taken so that Lip(del F) = Lip(G). We give a geometrical application of this result, concerning interpolation of sets by boundaries of C-1,C-1 convex bodies in H. Finally, we give a counterexample to a related question concerning smooth convex extensions of smooth convex functions with derivatives which are not uniformly continuous. (C) 2016 Elsevier Inc. All rights reserved.