Kirszbraun’s Theorem via an Explicit Formula

Abstract

Let \(X,Y\) be two Hilbert spaces, let E be a subset of \(X,\) and let \(G\colon E \to Y\) be a Lipschitz mapping. A famous theorem of Kirszbraun’s states that there exists \(\tilde {G} : X \to Y\) with \(\tilde {G}=G\) on E and \( \operatorname {\mathrm {Lip}}(\tilde {G})= \operatorname {\mathrm {Lip}}(G).\) In this note we show that in fact the function \(\tilde {G}:=\nabla _Y( \operatorname {\mathrm {conv}} (g))( \cdot , 0)\), where \[\begin{align*}g(x,y) = \inf_{z \in E} \Big\lbrace \langle G(z), y \rangle + \frac{\operatorname{\mathrm{Lip}}(G)}{2} \|(x-z,y)\|^2 \Big\rbrace + \frac{\operatorname{\mathrm{Lip}}(G)}{2}\|(x,y)\|^2, \end{align*}\]defines such an extension. We apply this formula to get an extension result for strongly biLipschitz mappings. Related to the latter, we also consider extensions of \(C^{1,1}\) strongly convex functions.