Geodesic
Geometry of the Gromov-Hausdorff distance on the class of all metric spaces
Sbornik. Mathematics, Tome 213 (2022) no. 5, pp. 641-658

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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. 
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S. I. Borzov; A. O. Ivanov; A. A. Tuzhilin. Geometry of the Gromov-Hausdorff distance on the class of all metric spaces. Sbornik. Mathematics, Tome 213 (2022) no. 5, pp. 641-658. http://geodesic.mathdoc.fr/item/SM_2022_213_5_a3/