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Article Dans Une Revue Canadian Mathematical Bulletin Année : 2021

Kirszbraun's theorem via an explicit formula

Résumé

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.

Dates et versions

hal-01975148 , version 1 (09-01-2019)

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Daniel Azagra, Erwan Le Gruyer, Carlos Mudarra. Kirszbraun's theorem via an explicit formula. Canadian Mathematical Bulletin, 2021, 64 (1), pp.142-153. ⟨10.4153/S0008439520000314⟩. ⟨hal-01975148⟩
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