Geodesic
Metric Segments in Gromov--Hausdorff class
Čebyševskij sbornik, Tome 23 (2022) no. 3, pp. 5-18.

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We study properties of metric segments in the class of all metric spaces considered up to an isometry, endowed with Gromov–Hausdorff distance. On the isometry classes of all compact metric spaces, the Gromov-Hausdorff distance is a metric. A metric segment is a class that consists of points lying between two given ones. By von Neumann–Bernays–Gödel (NBG) axiomatic set theory, a proper class is a “monster collection”, e.g., the collection of all sets. We prove that any metric segment in the proper class of isometry classes of all metric spaces with the Gromov-Hausdorff distance is a proper class if the segment contains at least one metric space at positive distances from the segment endpoints. If the distance between the segment endpoints is zero, then the metric segment is a set. In addition, we show that the restriction of a non-degenerated metric segment to compact metric spaces is a non-compact set.
Keywords: Gromov–Hausdorff distance, class of all metric spaces, von Neumann–Bernays–Gödel axioms, metric segment, compact set. 
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O. B. Borisova. Metric Segments in Gromov--Hausdorff class. Čebyševskij sbornik, Tome 23 (2022) no. 3, pp. 5-18. http://geodesic.mathdoc.fr/item/CHEB_2022_23_3_a0/

[1] Hausdorff F., Grundzüge der Mengenlehre, Veit, Leipzig, 1914 | MR | Zbl

[2] Gromov M., “Groups of Polynomial growth and Expanding Maps”, Publications Mathematiques Paris I.H.E.S., 53, 1981 | MR

[3] Burago D.Yu., Burago Yu.D., Ivanov S.V., Kurs metricheskoi geometrii, Izd-vo Instituta kompyuternykh issledovanii, M.–Izhevsk, 2004, 496 pp. | MR

[4] Edwards D., “The Structure of Superspace”, Studies in Topology, eds. Stavrakas N. M., Allen K. R., Academic Press, Inc, New York–London–San Francisco, 1975 | MR

[5] Tuzhilin A. A., Who Invented the Gromov–Hausdorff Distance?, 2017, arXiv: 1612.00728

[6] Mendelson E., Vvedenie v matematicheskuyu logiku, per. s angl. F.A. Kabakova pod red. S.I. Adyana, 2-oe iz., ispravlennoe, iz. «Nauka» gl. red. fiz-mat. lit., M., 1976, 320 pp.

[7] Ivanov A. O., Tuzhilin A. A., Geometriya rasstoyanii Khausdorfa i Gromova — Khausdorfa: sluchai kompaktov, Izd-vo Popechitelskogo soveta mekh-mat f-ta MGU, M., 2017, 111 pp.

[8] Ivanov A. O., Nikolaeva N. K., Tuzhilin A. A., “Metrika Gromova — Khausdorfa na prostranstve metricheskikh kompaktov - strogo vnutrennyaya”, Matem. zametki, 100:6 (2016), 947–950, arXiv: 1504.03830 | DOI | Zbl

[9] Ivanov A.O., Iliadis S., Tuzhilin A.A., Realizations of Gromov–Hausdorff Distance, 2016, arXiv: 1603.08850 | MR

[10] Ivanov A. O., Tuzhilin A. A., Hausdorff realization of linear geodesics of Gromov–Hausdorff space, 2019, arXiv: 1904.09281 | MR

[11] Ivanov A. O., Tuzhilin A. A., “Isometry group of Gromov–Hausdorff space”, Matematicki Vesnik, 71:1-2 (2019), 123–154 | MR | Zbl

[12] Memoli F., “On the Use of Gromov–Hausdorff Distances for Shape Comparison”, Proceedings of Point Based Graphics, eds. Botsch M., Pajarola R., Chen B., Zwicker M., The Eurographics Association, Prague, 2007, 81–90

[13] Klibus D.P., Kompaktnaya vypuklost sharov v prostranstve Gromova — Khausdorfa, Kursovaya rabota, Sait kafedry diff.geom. i prilozh. mekh-mata MGU, M., 2018 http://dfgm.math.msu.su/files/0students/2018-kr5-klibus.pdf

[14] John L. Kelley, General topology, D. van Nostrand Company, Inc., New York–Toronto–London, 1955 | MR | Zbl

[15] Engelking R., Obschaya topologiya, per. s angl. M. Ya. Antonovskogo i A.V. Arkhangelskogo, Mir, M., 1986, 752 pp. | MR