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

TL;DR

This paper characterizes when functions defined on arbitrary sets can be extended to convex functions with specified smoothness and normal conditions, providing new extension criteria and geometric applications.

Contribution

It offers necessary and sufficient conditions for extending functions to convex $C^{1, extomega}$ and $C^1$ convex functions with controlled derivatives, advancing convex extension theory.

Findings

Characterization of convex $C^{1, extomega}$ extensions with modulus control

Extension criteria for $C^1$ convex functions on compact sets

Geometric interpolation of convex bodies with prescribed normals

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, $\textrm{Lip}(F)\lesssim \|G\|_{\infty}$). 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 $K$.