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Zbl 1058.53061
Kirk, William A.
Geodesic geometry and fixed point theory. (English)
[A] Girela Álvarez, Daniel (ed.) et al., Seminar of mathematical analysis. Proceedings of the seminar which was held at the Universities of Malaga and Seville, Spain, September 2002--February 2003. Sevilla: Universidad de Sevilla, Secretariado de Publicaciones. 195-225 (2003). ISBN 84-472-0803-6/pbk

The author considers applications of some ideas of nonlinear analysis to $\Re_{0}$ domains of Aleksandrov spaces of curvature $\leq0$, often called CAT(0) spaces. An $\Re_{0}$ domain can be described as a convex domain in a metric space of curvature $\leq0$ in which minimizing geodesics depend continuously on their ends. For an arbitrary $K$, $\Re _{K}$ domains have been introduced by {\it A. D. Aleksandrov} in his seminar papers [A theorem on triangles in a metric space and some of its applications, Trudy Mat. Inst. Steklov 38, 5--23 (1951; Zbl 0049.39501) and Über eine Verallgemeinerung der Riemannschen Geometrie. Schr. Forschungsinst. Math. 1, 33--84 (1957; Zbl 0077.35702)]. In the first half of his paper, the author gives a survey of general properties of CAT$(K)$ domains mostly by following the book [{\it M. R. Bridson} and {\it A. Haefliger}, Metric spaces of non-positive curvature (1999; Zbl 0988.53001)]. At the beginning of the paper, the author explains the notation CAT(0) by writing that $C$, $A$ and $T$ stand for Cartan, Aleksandrov and Toponogov. Such explanation can mislead the reader. In fact, the definition of an $\Re_{K}$ domain (in terms of angle comparisons, as well as equivalent definitions in terms of $K$-concavity also called the CAT$(K) $ inequality) is entirely due to A. D. Aleksandrov. One of the most important properties of $\Re_{K}$ domains, that is used in the paper as definition of CAT$(K) $, is the property of $K$-concavity [{\it A. D. Aleksandrov}, Über eine Verallgemeinerung der Riemannschen Geometrie. Schr. Forschungsinst. Math. 1 (1957), p. 38], also known as CAT$(K) $ inequality. Another significant result used by the author is a generalization to $\Re_{K}$ domains of a familiar Busemann-Feller theorem: The projection to a convex set is a nonexpansive mapping (for the first application to $\Re_{K}$ domains, see [{\it I. G. Nikolaev}, Sib. Math. J. 20, 246--252 (1979; Zbl 0434.53045)]. In the second part of his work, the author presents results connected with the fixed point theory in $\Re_{0}$ domains. One of the author's results states that if $K$ is a bounded closed convex subset in a complete CAT$(0) $ space and $\ f$ is a nonexpansive mapping with the property $\inf\{d(x,f(x)): x\in K\} =0$, then $f$ has a fixed point in $K$. Among the results presented are those connected with the approximate fixed point property and homotopy invariance theorems. These results are closely connected to theorems 23.1, 32.3 and 24.1 in [{\it K. Goebel} and {\it S. Reich} Uniform convexity, hyperbolic geometry, and nonexpansive mappings (1984; Zbl 0537.46001)]. Applications to graph theory are given.
[I. G. Nikolaev (Urbana)]

MSC 2000:: *53C70 Direct methods (G-spaces of Busemann, etc.)
53C45 Global surface theory (a la A.D. Aleksandrov)
58C30 Fixed point theorems on manifolds
05C75 Structural characterization of types of graphs

Keywords: metric spaces; convex sets; CAT(k) spaces; nonexpansive mappings; fixed point theory; graphs

Citations: Zbl 0049.39501; Zbl 0988.53001; Zbl 0434.53045; Zbl 0537.46001; Zbl 0077.35702

Cited in: Zbl 1216.47085 Zbl 1083.53061

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