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

Abstract

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 $\text{Lip}(F)\lesssim \Vert G{\Vert }_{\infty }$. 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 ${\mathbb{R}}^n$.

Final version. We have added an example and some new comments to the introduction

Cited by in corpus (1)

• Monotone approximation of differentiable convex functions with applications to general minimization problems