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

Original · EN

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.