Smooth convex extensions of convex functions

TL;DR

This paper investigates conditions under which convex functions defined on convex sets can be extended smoothly and convexly to the entire space, providing solutions especially for the case of infinite smoothness.

Contribution

It offers a comprehensive characterization for smooth convex extensions in the case of infinite differentiability and near-optimal results for finite smoothness, especially for intersections of ovaloids.

Findings

Solved the problem for $m=\infty$ case.

Provided partial solutions for finite $m\geq 2$.

Achieved near-optimal results for intersections of ovaloids.

Abstract

Let $C$ be a compact convex subset of $\mathbb{R}^n$, $f:C\to\mathbb{R}$ be a convex function, and $m\in\{1, 2, ..., \infty\}$. Assume that, along with $f$, we are given a family of polynomials satisfying Whitney's extension condition for $C^m$, and thus that there exists $F\in C^{m}(\mathbb{R}^n)$ such that $F=f$ on $C$. It is natural to ask for further (necessary and sufficient) conditions on this family of polynomials which ensure that $F$ can be taken to be convex as well. We give a satisfactory solution to this problem in the case $m=\infty$, and also less satisfactory solutions in the case of finite $m\geq 2$ (nonetheless obtaining an almost optimal result for $C$ a finite intersection of ovaloids). For a solution to a similar problem in the case $m=1$ (even for $C$ not necessarily convex), see arXiv:1507.03931, arXiv:1706.09808, arXiv:1706.02235.