Differentiable convex extensions with sharp Lipschitz constants

TL;DR

This paper characterizes when convex functions with certain smoothness can be extended from subsets of superreflexive Banach spaces to the whole space with optimal Lipschitz constants, providing explicit formulas and sharp estimates.

Contribution

It provides a complete characterization and explicit formulas for convex extensions with sharp Lipschitz constants in superreflexive Banach spaces, including Hilbert spaces.

Findings

Extensions have sharp Lipschitz constants equal to the supremum norm of the gradient.

Explicit formulas for convex extensions are derived.

Sharp estimates for the growth of the seminorms are obtained, especially in Hilbert spaces.

Abstract

Given a superreflexive Banach space $X$, and a set $E \subset X$, we characterise the $1$-jets $(f,G)$ on $E$ that admit $C^{1,\omega}$ convex extensions $(F,DF)$ to all of $X$; where $\omega$ is any admissible modulus of continuity depending on the regularity of $X$. Moreover, we obtain precise estimates for the growth of the $C^{1,\omega}$ seminorm of the extensions with respect to the initial data. We show how these estimates can be improved in the Hilbert setting, and are asymptotically sharp for H\"older moduli. Remarkably, our extensions have the sharp Lipschitz constant $\mathrm{Lip}(F,X) = \|G\|_{L^\infty(E)}$, when $G$ is a bounded map. All these extensions are given by simple and explicit formulas. We also prove a similar theorem for $C^1$ convex extensions of jets defined on compact subsets $E$ of superreflexive spaces $X$, with the sharp Lipschitz constant too. The results are new even when $X=\mathbb{R}^n.$