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: $d_{GH}\bigl(\mathbb{Z}^n,\,\lambda\mathbb{Z}^n\bigr)\ge\frac{1}{2}$. 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.