Kirszbraun's theorem via an explicit formula

2018/10/24 by Azagra, Daniel, Gruyer, Erwan Le, Mudarra, Carlos · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1810.10288

Abstract

Let X,Y be two Hilbert spaces, E a subset of X and G: E → Y a Lipschitz mapping. A famous theorem of Kirszbraun's states that there exists \widetildeG : X → Y with \widetildeG=G on E and \textrmLip(\widetildeG)=\textrmLip(G). In this note we show that in fact the function \widetildeG:=∇_Y(\textrmconv(g))( ⋅ , 0), where g(x,y) = inf_{z ∈ E} \lbrace ⟨ G(z), y ⟩ + \tfracM2 ‖(x-z,y)‖² \rbrace + \tfracM2‖(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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