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

Abstract

In the paper we prove that, for arbitrary unbounded subset $A\subset R$ and an arbitrary bounded metric space~$X$, a curve $A{\times }_{{\ell }^1}(tX)$, $t\in [0,\infty )$ is a geodesic line in the Gromov--Hausdorff class. We also show that, for abitrary $\lambda >1$, $n\in \mathbb{N}$, the following inequality holds: $d_{GH}({\mathbb{Z}}^n,\lambda {\mathbb{Z}}^n)\ge \frac{1}{2}$. We conclude that a curve $t{\mathbb{Z}}^n$, $t\in (0,\infty )$ 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 ${\mathbb{R}}^n$ on some~$\lambda >0$ is also discontinous with respect to the Gromov--Hausdorff distance.

9 pages, 2 pictures, accepted for publication in Chebyshevskii Sbornik