An Extension Theorem for convex functions of class $C^{1,1}$ on Hilbert spaces

Abstract

Let $\mathbb{H}$ be a Hilbert space, $E\subset \mathbb{H}$ be an arbitrary subset and $f:E\to \mathbb{R},G:E\to \mathbb{H}$ 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 $\text{Lip}(\nabla F)=\text{Lip}(G)$. 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.

In this new version we provide an application of the main result concerning interpolation of sets by boundaries of convex bodies. We also give a counterexample of a related question concerning extensions of smooth convex functions with derivatives which are not uniformly continuous