Whitney Extension Theorems for convex functions of the classes $C^1$ and $C^{1,\omega }$

Abstract

Let $C$ be a subset of ${\mathbb{R}}^n$ (not necessarily convex), $f:C\to \mathbb{R}$ be a function, and $G:C\to {\mathbb{R}}^n$ be a uniformly continuous function, with modulus of continuity $\omega$. We provide a necessary and sufficient condition on $f$, $G$ for the existence of a convex function $F\in C^{1,\omega }({\mathbb{R}}^n)$ such that $F=f$ on $C$ and $\nabla F=G$ on $C$, with a good control of the modulus of continuity of $\nabla F$ in terms of that of $G$. On the other hand, assuming that $C$ is compact, we also solve a similar problem for the class of $C^1$ convex functions on ${\mathbb{R}}^n$, with a good control of the Lipschitz constants of the extensions (namely, $\text{Lip}(F)\lesssim \Vert G{\Vert }_{\infty }$). Finally, we give a geometrical application concerning interpolation of compact subsets $K$ of ${\mathbb{R}}^n$ by boundaries of $C^1$ or $C^{1,1}$ convex bodies with prescribed outer normals on $K$.

arXiv commentSome additional comments and improvements suggested by a referee have been added. arXiv admin note: text overlap with arXiv:1501.05226