This article is cited in 3 papers

Geometry of the Gromov-Hausdorff distance on the class of all metric spaces

S. I. Borzov^a, A. O. Ivanov^bcd, A. A. Tuzhilin<sup>b

a</sup> Pyrus JSC, Moscow, Russia
^b Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^c Bauman Moscow State Technical University, Moscow, Russia
^d Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia

Abstract: The geometry of the Gromov-Hausdorff distance on the class of all metric spaces considered up to isometry is studied. The concept of a class in the sense of von Neumann-Bernays-Gödel set theory is used. As in the case of compact metric spaces, continuous curves and their lengths are defined, and the Gromov-Hausdorff distance is shown to be intrinsic on the entire class. As an application, metric segments (classes of points lying between two given points) are considered and their extendability beyond endpoints is examined.
Bibliography: 13 titles.

Keywords: Gromov-Hausdorff distance, class of all metric spaces, geodesic, metric segment, extendability of a geodesic.

MSC: 51F99, 51K05

Received: 09.08.2021

DOI: 10.4213/sm9651

 English version: Sbornik: Mathematics, 2022, 213:5, 641–658

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