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 domains of Aleksandrov spaces of curvature , often called CAT(0) spaces. An domain can be described as a convex domain in a metric space of curvature in which minimizing geodesics depend continuously on their ends. For an arbitrary , 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 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 , and stand for Cartan, Aleksandrov and Toponogov. Such explanation can mislead the reader. In fact, the definition of an domain (in terms of angle comparisons, as well as equivalent definitions in terms of -concavity also called the CAT inequality) is entirely due to A. D. Aleksandrov. One of the most important properties of domains, that is used in the paper as definition of CAT, is the property of -concavity [A. D. Aleksandrov, Über eine Verallgemeinerung der Riemannschen Geometrie. Schr. Forschungsinst. Math. 1 (1957), p. 38], also known as CAT inequality. Another significant result used by the author is a generalization to domains of a familiar Busemann-Feller theorem: The projection to a convex set is a nonexpansive mapping (for the first application to 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 domains. One of the author’s results states that if is a bounded closed convex subset in a complete CAT space and is a nonexpansive mapping with the property , then has a fixed point in . 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.