*(English)*Zbl 1058.53061

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 $inf\left\{d\right(x,f\left(x\right)):x\in K\}=0$, 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 [*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 |