Geodesic
The Gromov--Hausdorff distances to simplexes
Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 108-122.

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In the paper geometrical characteristics of metric spaces appearing in explicit formulas for the Gromov–Hausdorff distance from this spaces to so-called simplexes, i.e., the metric spaces, all whose non-zero distances are the same. For the calculation of those distances the geometry of partitions of these spaces is important. In the case of finite metric spaces that leads to some analogues of the edges lengths of minimal spanning trees. Earlier, a similar theory was elaborated for compact metric spaces. These results are generalised to the case of an arbitrary bounded metric space, explicit formulas are obtained, and some proofs are simplified.
Keywords: Gromov–Hausdorff distance, metric geometry, metric space. 
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D. S. Grigor'ev; A. O. Ivanov; A. A. Tuzhilin. The Gromov--Hausdorff distances to simplexes. Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 108-122. http://geodesic.mathdoc.fr/item/CHEB_2019_20_2_a7/

[1] F. Hausdorff, Grundzüge der Mengenlehre, Veit, Leipzig ; reprinted Chelsea, 1949 | MR | Zbl

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

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

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

[5] Burago D. Yu., Burago Yu. D., Ivanov S. V., A Course in Metric Geometry, American Mathematical Society, Providence, RI, 2001 | MR | MR | Zbl

[6] Ivanov A. O., Tuzhilin A. A., Geometry of Hausdorff and Gromov–Hausdorff distances, the case of compact spaces, Izd-vo Popech. Soveta Mech. Mat. Facult. MGU, M., 2017 (in Russian)

[7] Ivanov A. O., Nikolaeva N. K., Tuzhilin A. A., “The Gromov–Hausdorff Metric on the Space of Compact Metric Spaces is Strictly Intrinsic”, Mathematical Notes, 100:6 (2016), 171–173 | MR | Zbl

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

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

[10] A. A. Tuzhilin, Calculation of Minimum Spanning Tree Edges Lengths using Gromov–Hausdorff Distance, 2016, arXiv: 1605.01566

[11] S. D. Iliadis, A. O. Ivanov, A. A. Tuzhilin, Geometry of Compact Metric Space in Terms of Gromov-Hausdorff Distances to Regular Simplexes, 2016, arXiv: 1607.06655 | MR

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

[13] A. O. Ivanov, A. A. Tuzhilin, Gromov–Hausdorff Distance, Irreducible Correspondences, Steiner Problem, and Minimal Fillings, 2016, arXiv: 1604.06116

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

[15] A. Ivanov, A. Tuzhilin, “Geometry of Gromov–Hausdorff metric space”, Bulletin de l'Academie Internationale CONCORDE, 2017, no. 3, 47–57 | MR