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

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