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Kirszbraun’s Theorem via an Explicit Formula

Part of: Nonlinear operators and their properties Maps and general types of spaces defined by maps General convexity

Published online by Cambridge University Press:  29 April 2020

Daniel Azagra ,
Erwan Le Gruyer  and
Carlos Mudarra
Daniel Azagra*
Affiliation:
ICMAT (CSIC-UAM-UC3-UCM), Departamento de Análisis Matemático y Matemática Aplicada, Facultad Ciencias Matemáticas, Universidad Complutense, 28040, Madrid, Spain
Erwan Le Gruyer
Affiliation:
INSA de Rennes & IRMAR, 20, Avenue des Buttes de Coësmes, CS 70839 F-35708, Rennes Cedex 7, France e-mail: Erwan.Le-Gruyer@insa-rennes.fr
Carlos Mudarra
Affiliation:
ICMAT (CSIC-UAM-UC3-UCM), Calle Nicolás Cabrera 13-15. 28049Madrid, Spain e-mail: carlos.mudarra@icmat.es
*
e-mail: azagra@mat.ucm.es
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Abstract

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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.

Keywords

Lipschitz functionKirszbraun theorem

MSC classification

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc.
Secondary: 52A41: Convex functions and convex programs 54C20: Extension of maps
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 142 - 153
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2020

Footnotes

D. Azagra and C. Mudarra were partially supported by Grant MTM2015-65825-P and by the Severo Ochoa Program for Centres of Excellence in R & D (Grant SEV-2015-0554).

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