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Geodesic geometry and fixed point theory. (English) Zbl 1058.53061
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 (ISBN 84-472-0803-6/pbk). 195-225 (2003).

The author considers applications of some ideas of nonlinear analysis to ${\Re }_{0}$ domains of Aleksandrov spaces of curvature $\le 0$, often called CAT(0) spaces. An ${\Re }_{0}$ domain can be described as a convex domain in a metric space of curvature $\le 0$ in which minimizing geodesics depend continuously on their ends. For an arbitrary $K$, ${\Re }_{K}$ domains have been introduced by 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$\left(K\right)$ domains mostly by following the book [M. R. Bridson and 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$\left(K\right)$ 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$\left(K\right)$, is the property of $K$-concavity [A. D. Aleksandrov, Über eine Verallgemeinerung der Riemannschen Geometrie. Schr. Forschungsinst. Math. 1 (1957), p. 38], also known as CAT$\left(K\right)$ 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 [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$\left(0\right)$ space and $\phantom{\rule{4pt}{0ex}}f$ is a nonexpansive mapping with the property $inf\left\{d\left(x,f\left(x\right)\right):x\in K\right\}=0$, 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 [K. Goebel and S. Reich Uniform convexity, hyperbolic geometry, and nonexpansive mappings (1984; Zbl 0537.46001)]. Applications to graph theory are given.

##### MSC:
 53C70 Direct methods ($G$-spaces of Busemann, etc.) 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov) 58C30 Fixed point theorems on manifolds 05C75 Structural characterization of families of graphs