Convex C-1 extensions of 1-jets from compact subsets of Hilbert spaces

Let X denote a Hilbert space. Given a compact subset K of X and two continuous functions f : K -> R., G : K -> X, we show that a necessary and sufficient condition for the existence of a convex function F epsilon C-1 (X) such that F = f on K and del F = G on K is that the 1-jet (f, G) satisfies:

(1) f(x) >= f(y) + <G(y), x - y > for all x, y epsilon K, and

(2) if x, y epsilon K and f(x) = f(y) + <G(y), x - y > then G(x) =G(y).

We also solve a similar problem for K replaced with an arbitrary bounded subset of X, and for C-1 (X) replaced with the class C-b(1,u) (X) of differentiable functions with uniformly continuous derivatives on bounded subsets of X.