Global geometry and $C^1$ convex extensions of 1-jets

Abstract

Let $E$ be an arbitrary subset of ${ℝ}^n$ (not necessarily bounded) and $f:E\to ℝ$, $G:E\to {ℝ}^n$ be functions. We provide necessary and sufficient conditions for the $1$-jet $\left(f,G\right)$ to have an extension $\left(F,\nabla F\right)$ with $F:{ℝ}^n\to ℝ$ convex and $C^1$. Additionally, if $G$ is bounded we can take $F$ so that $Lip\left(F\right)\lesssim \parallel G{\parallel }_{\infty }$. 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 ${ℝ}^n$.