Let $E$ be an arbitrary subset of $\mathbb{ R}^n$ (not necessarily bounded), and $f:E\to\mathbb{ R}$, $G:E\to\mathbb{ R}^n$ be functions. We provide necessary and sufficient conditions for the $1$-jet $(f,G)$ to have an extension $(F, \nabla F)$ with $F:\mathbb{ R}^n\to\mathbb{ R}$ convex and of class $C^{ 1}$. Besides, if $G$ is bounded we can take $F$ so that $\textrm{ Lip}(F)\lesssim \|G\|_{ \infty}$. As an application we also solve a similar problem about finding convex hypersurfaces of class $C^1$ with prescribed normals at the points of an arbitrary subset of $\mathbb{ R}^n$.
Categories: math.DG math.FA