Kirszbraun's theorem and metric spaces of bounded curvature
type: Item
instance of: scholarly article
title: Kirszbraun's theorem and metric spaces of bounded curvature
zbMATH Open document ID: 0891.53046
published in: Geometric and Functional Analysis. GAFA
publication date: 1998-07-06T00:00:00Z
review text: It is known that every Lipschitz map \(f: S\to\mathbb R^n\) with \(S\subseteq \mathbb R^m\) admits a Lipschitz extension \(\overline{f}:{\mathbb R}^{m}\rightarrow {\mathbb R}^n\) with the same Lipschitz constant (Theorem of Kirszbraun) [\textit{M. D. Kirszbraun}, Fundam. Math. 22, 77--108 (1934; Zbl 0009.03904)]. For a ``short and readable proof'', according to the authors, see \textit{H. Federer} [Geometric measure theory. Berlin etc.: Springer-Verlag (1969; Zbl 0176.00801)]. In the present work the authors prove the following generalization: Theorem A (generalized Kirszbraun theorem): Let \(k\in {\mathbb R}\) , and let \(X\) and \(Y\) be two geodesic spaces, such that all triangles of perimeter \(<2D_k\) in \(X\) and \(Y\) are \(k\)-thick or \(k\)-thin, respectively. Assume that \(Y\) is complete. Let \(S\) be an arbitrary subset of \(X\), and \(f: S\rightarrow Y\) a 1-Lipschitz map with \(\operatorname{diam} f(S)\leq D_k/2\), then there exists a 1-Lipschitz extension \(\overline{f}: X\rightarrow Y\) of \(f\). The number \(D_k\) is the diameter of the model surface of constant Gauss curvature \(k\). The motivation for this problem comes from the study of quasiflats in Hadamard spaces [\textit{U. Lang} and \textit{V. Schroeder}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 30, 339--352 (1997; Zbl 0876.53050)]. A Hadamard space in the sense of Alexandrov is a complete metric space which is non-positively curved in the sense of Alexandrov. The authors give an introduction to the theory of Alexandrov spaces and provide a large bibliography for more details. The main argument used to prove this theorem is to show that every 1-Lipschitz map \(f:E\rightarrow Y\) defined on a finite subset \(E=\{x_1,\ldots,x_n\} \) of \(X\) can be extended to an additional point \(x\in X\). This can be done if one considers the set \(A(c):= \{y\in Y: d(y,f(x_i))\leq cd(x,x_{i})\) for \(i=1,\ldots n\} \) for some constant factor \(c\). If \(c=\inf_{b\geq 0}A(b)\neq \emptyset \), then \(c\leq 1\), …
reviewed by: Ana Pereira do Vale
Mathematics Subject Classification ID: 53C45
Mathematics Subject Classification ID: 26A16
zbMATH DE Number: 1054109
zbMATH Keywords: Lipschitz maps
zbMATH Keywords: bounded curvature
zbMATH Keywords: Alexandrov spaces
zbMATH Keywords: thin and thick triangles
zbMATH Keywords: tangent cones
author: Urs Lang
author: Viktor Schroeder
MaRDI profile type: Publication
full work available at URL: s000390050018
OpenAlex ID: W2157321806
DOI: 10.1007/S000390050018
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description: scientific article; zbMATH DE number 1054109