Kirszbraun's theorem via an explicit formula

TL;DR

This paper presents an explicit formula for extending Lipschitz functions between Hilbert spaces, providing a constructive approach to Kirszbraun's theorem and applications to biLipschitz homeomorphisms and convex functions.

Contribution

It introduces a new explicit formula for Kirszbraun extensions using convex analysis and gradients, enhancing the constructive methods for Lipschitz extensions.

Findings

Provides an explicit formula for Lipschitz extension in Hilbert spaces.

Applies the formula to extend strongly biLipschitz homeomorphisms.

Discusses extensions of $C^{1,1}$ strongly convex functions.

Abstract

Let $X,Y$ be two Hilbert spaces, $E$ a subset of $X$ and $G: E \to Y$ a Lipschitz mapping. A famous theorem of Kirszbraun's states that there exists $\widetilde{G} : X \to Y$ with $\widetilde{G}=G$ on $E$ and $\textrm{Lip}(\widetilde{G})=\textrm{Lip}(G).$ In this note we show that in fact the function $$\widetilde{G}:=\nabla_Y(\textrm{conv}(g))( \cdot , 0), \qquad \text{where} $$ $$ g(x,y) = \inf_{z \in E} \lbrace \langle G(z), y \rangle + \tfrac{M}{2} \|(x-z,y)\|^2 \rbrace + \tfrac{M}{2}\|(x,y)\|^2, $$ defines such an extension. We apply this formula to get an extension result for {\em strongly biLipschitz homeomorphisms.} Related to the latter, we also consider extensions of $C^{1,1}$ strongly convex functions.