The boundary of a quasiconvex space
\textit{G. R. Conner} [Lond. Math. Soc. Lect. Note Ser. 275, 28-38 (2000; Zbl 0986.20040)] has defined a geodesic metric space to be quasiconvex if for some \(\delta > 0\) the distance between the midpoints of any two sides of any geodesic triangle is no more than half the length of the third side plus \(\delta\). Quasiconvex metric spaces include two important classes of metric spaces: hyperbolic metric spaces in the sense of Gromov and CAT(0) metric spaces. For a proper (i.e., every closed ball is compact) quasiconvex space \(X\), the author defines the boundary \(\partial X\) to be the set of asymptotic equivalence classes of geodesic rays in \(X\) and topologizes \(\bar X = X\cup \partial X\) so that \(\bar X\) is a compact metric space containing \(X\) as a dense open subspace. The construction specializes to the Gromov boundary for hyperbolic metric spaces and to the visual boundary for CAT(0) metric spaces (with the cone topology on \(X\cup \partial X\)), providing a nice unification of these cases. \textit{M. R. Bridson} and \textit{A. Haefliger} [Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften 319, Springer, Berlin (1999; Zbl 0988.53001)] contains a definitive treatment of the special cases.
scientific article; zbMATH DE number 1421588