Abstract:
In the paper we prove that, for arbitrary unbounded subset $A\subset \mathbb{R}$ 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: $\mathrm{dist}_{GH}\bigl(\mathbb{Z}^n, \lambda\mathbb{Z}^n\bigr)\ge\frac{1}{2}$. 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.
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