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

Let HH be a Hilbert space, E⊂HE⊂H be an arbitrary subset and f:E→Rf:E→R, G:E→HG:E→H be two functions. We give a necessary and sufficient condition on the pair (f,G)(f,G) for the existence of a convex   function F∈C1,1(H)F∈C1,1(H) such that F=fF=f and ∇F=G∇F=G on E. We also show that, if this condition is met, F   can be taken so that Lip(∇F)=Lip(G). We give a geometrical application of this result, concerning interpolation of sets by boundaries of C1,1C1,1 convex bodies in HH. Finally, we give a counterexample to a related question concerning smooth convex extensions of smooth convex functions with derivatives which are not uniformly continuous.