Locally $C^{1,1}$ convex extensions of $1$-jets

Let $E$ be an arbitrary subset of ${\mathbb{R}}^n$, and $f:E\to \mathbb{R}$, $G:E\to {\mathbb{R}}^n$ be given functions. We provide necessary and sufficient conditions for the existence of a convex function $F\in C^{1,1}_{$\text{loc}$}($\mathbb{R}$^n)$suchthat$F=f$and$\nabla F=G$on$E$. We give a useful explicit formula for such an extension $F$, and a variant of our main result for the class $C_{\text{loc}}^{1,\omega }$, where $\omega$ is a modulus of continuity. We also present two applications of these results, concerning how to find $C_{\text{loc}}^{1,1}$ convex hypersurfaces with prescribed tangent hyperplanes on a given subset of ${\mathbb{R}}^n$, and some explicit formulas for (not necessarily convex) $C_{\text{loc}}^{1,1}$ extensions of $1$-jets.