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Kirszbraun's theorem via an explicit formula (1810.10288v3)

Published 24 Oct 2018 in math.FA

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.

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