  Let $\mathbb{H}$ be a Hilbert space, $E \subset \mathbb{H}$ be an arbitrary
subset and $f: E \rightarrow \mathbb{R}, \: G: E \rightarrow \mathbb{H}$ be two
functions. We give a necessary and sufficient condition on the pair $(f,G)$ for
the existence of a \textit{convex} function $F\in C^{1,1}(\mathbb{H})$ such
that $F=f$ and $\nabla F =G$ on $E$. We also show that, if this condition is
met, $F$ can be taken so that $\textrm{Lip}(\nabla F) = \textrm{Lip}(G)$. We
give a geometrical application of this result, concerning interpolation of sets
by boundaries of $C^{1,1}$ convex bodies in $\mathbb{H}$. Finally, we give a
counterexample to a related question concerning smooth convex extensions of
smooth convex functions with derivatives which are not uniformly continuous.
