Kirszbraun's theorem and metric spaces of bounded curvature
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 \(c\leq 1\), and \(A(c)\) contains one single element which is the candidate to \(\overline{f}(x).\) In order to prove the details of the procedure explained above, the authors developed new tools in the theory of tangent cones of these spaces. These results are explained in Section 3. The article also includes two other theorems related to the generalization of the theorem of Kirszbraun. The first one, Theorem B (Lipschitz maps for trees), states the following: Let \(X\) be an arbitrary metric space and \(Y\) a complete geodesic metric space such that every triangle in \(Y\) is \(k\)-thin for all \(k\leq 0\). Let \(S\) be an arbitrary subset of \(X\) and \(f:S\rightarrow Y\) a Lipschitz map with constant \(\text{Lip } f\). Then there exists a Lipschitz extension \(\overline{f}:X\rightarrow Y\) of \(f\) with \(\text{Lip } \overline{f}=\text{Lip } f\). This theorem can be considered as the limit case of Theorem A as \(k=-\infty \) and is a known result for \(Y=\mathbb{R}\) ([\textit{E. J. McShane}, Bull. Am. Math. Soc. 40, 837--842 (1934; Zbl 0010.34606)] and [\textit{M. Gromov}, `Asymptotic invariants of infinite groups', in: ``Geometric group theory'', Vol. 2. Lond. Math. Soc. Lect. Notes, Ser. 182. Cambridge: Cambridge University Press (1993; Zbl 0841.20039)]). The other theorem states the fact that if the initial set where the map is defined consists of three points, the curvature condition on \(X\) in Theorem A can be weakened. Theorem C (extensions to quadruples): Let \(k\in\mathbb{R}\), \(X\) be a metric space with all quadruples in \(X\) satisfying the \(\gamma ^{k}\) condition and \(Y\) be a geodesic metric space such that all triangles in \(Y\) of perimeter \(<2D_{k}\) are \(k\)-thin; whenever \(x_{1},\dots,x_{4}\in X\) and \(f: \{x_1,x_2,x_3\} \rightarrow Y\) is a 1-Lipschitz map whose image has diameter \(D_{k}/2\) , then there exists a 1-Lipschitz extension \(\overline{f}: \{x_1,x_2,x_3,x_4\} \rightarrow Y\) of \(f\). A quadruple \((x; x_1,x_2,x_3)\) in \(X\) is said to satisfy the \(\gamma^k\) condition if the three angles with base point \(x\) have comparison angles in the model surface of constant Gauss curvature \(k\) whose measures add up to at most \(2\pi\). The authors mention other generalizations of the Theorem of Kirszbraun and list some bibliography on these generalizations. For more comments on the theorem as well as some of its history, the authors refer to \textit{L. Danzer}, \textit{B. Grünbaum} and \textit{V. Klee} [Proc. Symp. Pure Math. 7, 101--180 (1963; Zbl 0132.17401)].
scientific article; zbMATH DE number 1054109