The sharp Whitney extension theorem for convex $C^1$ Lipschitz functions

TL;DR

This paper develops a method to extend convex functions defined on arbitrary sets in Euclidean space to the entire space, preserving convexity, differentiability, and Lipschitz constants, with control over their global behavior.

Contribution

It provides a sharp Whitney extension theorem for convex $C^1$ Lipschitz functions, establishing necessary and sufficient conditions for such extensions and constructing them explicitly.

Findings

Extensions preserve Lipschitz constants exactly.

Extensions can be tailored with prescribed coercivity directions.

The method applies to arbitrary sets in Euclidean space.

Abstract

For an arbitrary set $E \subset \mathbb{R}^n$, and functions $f:E \to \mathbb{R}$, $G: E\to \mathbb{R}^n$ with $G$ bounded, we construct $C^1(\mathbb{R}^n)$ convex extensions $(F, \nabla F)$ of $(f,G)$ with the sharp Lipschitz constant $$ \mathrm{Lip}(F) = \sup_{x\in E} |G(x)|, $$ provided that $(f,G)$ satisfies the pertinent necessary and sufficient conditions for $C^1$ convex, and Lipschitz extendability. Also, these extensions can be constructed with prescribed global behavior in terms of directions of coercivity.