New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line

2025/04/20 by Mikhailov, Ivan N.
#51F99 #53C22 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2504.14629

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: d_GH(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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