New geodesic lines in the Gromov – Hausdorff class lying in the cloud of the real line
https://doi.org/10.22405/2226-8383-2025-26-2-176-185
Abstract
In the paper we prove that, for arbitrary unbounded subset 𝐴 ⊂ 𝑅 and an arbitrary bounded metric space 𝑋, a curve 𝐴 ×ℓ1 (𝑡𝑋), 𝑡 ∈ [0, ∞) is a geodesic line in the Gromov – Hausdorff class. We also show that, for abitrary 𝜆 > 1, 𝑛 ∈ N, the following inequality holds: dist𝐺𝐻 (︀Z𝑛, 𝜆Z𝑛)︀⩾ 1/2 . We conclude that a curve 𝑡Z𝑛, 𝑡 ∈ (0, ∞) is not continuous with respect to the Gromov – Hausdorff distance, and, therefore, is not a gedesic line. Moreover, it follows that multiplication of all metric spaces lying on the finite Gromov – Hausdorff distance from R𝑛
on some 𝜆 > 0 is also discontinous with respect to the Gromov – Hausdorff distance.
References
1. Bogaty, S.A., Tuzhilin, A.A., 2021, “Gromov – Hausdorff class: its completeness and cloud geometry”, ArXiv e-prints, arXiv:2110.06101.
2. Burago, D., Burago, Yu., Ivanov, S., 2001, A Course in Metric Geometry, Graduate Studies in Mathematics 33, AMS.
3. Edwards, D., 1975, “The structure of superspace”, in Studies in Topology, Academic Press.
4. Gromov, M., 1981, “Groups of polynomial growth and expanding maps”, Publications Mathematiques I.H.E.S., vol. 53, pp. 53–78.
5. Gromov, M., 1981, Structures m´etriques pour les vari´et´es riemanniennes, Textes Math. 1.
6. Carlsson, G.E., Memoli, F., 2010, “Characterization, stability and convergence of hierarchical clustering methods”, J. Mach. Learn., vol. 11, no. 47, pp. 1425–1470.
7. Gromov, M., 1999, Metric structures for Riemannian and non-Riemannian spaces, Birkh¨auser.
8. Kang, H., Sobolev, A.V., 2010, “Distribution of integer lattice points in a ball centred at a diophantine point”, Mathematika, vol. 56, no. 1, pp. 118–134.
9. Lim, S., Memoli, F., Smith, Z., 2022, “The Gromov – Hausdorff distance between spheres”, ArXiv e-prints, arXiv:2105.00611v5.
10. Tuzhilin, A.A., 2019, Lectures on Hausdorff and Gromov – Hausdorff distance geometry, arXiv:2012.00756.
11. Tuzhilin, A.A., 2016, “Who invented the Gromov – Hausdorff Distance?”, ArXiv e-prints, arXiv:1612.00728.
12. Vihrov, A., 2024, “Denseness of metric spaces in general position in the Gromov – Hausdorff class”, Topol. Its Appl., vol. 342, 108771.
Review
For citations:
Mikhailov I.N. New geodesic lines in the Gromov – Hausdorff class lying in the cloud of the real line. Chebyshevskii Sbornik. 2025;26(2):176-185. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-2-176-185
Views:
9
Email this article 