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.
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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
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