Kirszbraun's theorem via an explicit formula

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 $\tilde{G}:X\to Y$ with $\tilde{G}=G$ on $E$ and $\text{Lip}(\tilde{G})=\text{Lip}(G).$ In this note we show that in fact the function $$\tilde{G}:={\nabla }_Y(\text{conv}(g))(\cdot ,0),\text{where}$$ $$g(x,y)=\underset{z\in E}{\inf }\{⟨G(z),y⟩+\frac{M}{2}\Vert (x-z,y){\Vert }^2\}+\frac{M}{2}\Vert (x,y){\Vert }^2,$$ 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.

9 pages. We have added some comments and explanations