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Ivan N. Mikhailov

Publications and source records attributed to Ivan N. Mikhailov.

6 recordsLinked to original sources
arXiv2026

Gromov-Hausdorff distance and Jung constant of finite-dimensional normed spaces

For a finite-dimensional normed space $V$ and a subset $X$ with finite Hausdorff distance from $V$, we prove that the Gromov--Hausdorff distance between $X$ and $V$ is at least the Hausdorff distance between $X$ and $V$, divided by twice the relative Jung constant of $V$. If $V$ furthermore satisfies a certain intersection property, we show a stronger result where the relative Jung constant can be replaced with its absolute version. Key words: Normed spaces, Jung constant, Hausdorff distance, Gromov--Hausdorff distance.

math.MG↗
arXiv2025

Lipschitz distance between clouds

In this note we show that the Lipschitz distance between the classes of metric spaces at finite Gromov-Hausdorff distances from the one-point metric space and the real line with the natural metric, respectively, is positive.

math.MG↗
arXiv2025

Calculating Gromov-Hausdorff distance by means of asymptotic dimension

In this paper, we apply the concept of asymptotic dimension to calculating Gromov-Hausdorff distances between some unbounded metric spaces. For example, we show that the Gromov--Hausdorff between $\mathbb{R}^2$ with the Euclidean metric and $\mathbb{Z}^2$ equals the Hausdorff distance between them: $d_{GH}(\mathbb{R}^2, \mathbb{Z}^2) = d_H(\mathbb{R}^2, \mathbb{Z}^2)$.

math.MG↗
arXiv2025

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

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 $λ> 1$, $n\in\mathbb{N}$, the following inequality holds: $d_{GH}\bigl(\mathbb{Z}^n,\,λ\mathbb{Z}^n\bigr)\ge\frac{1}{2}$. 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~$λ> 0$ is also discontinous with respect to the Gromov--Hausdorff distance.

math.MG↗