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

Abstract

Let H be a Hilbert space, E⊂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∈C1,1(H) such that F=f and ∇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 C1,1 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.