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

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 $$\mathrm{Lip}(F)=\underset{x\in E}{\sup }|G(x)|,$$ 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.

17 pages