Searcharxiv⌕ Search

arXiv subjects

Alexey Tuzhilin

Publications and source records attributed to Alexey Tuzhilin.

12 recordsLinked to original sources
arXiv2025

Curves in hyperspaces obtained by intersection of $r$-neighborhoods with a fixed subset

The present paper generalizes the result from one of the papers by Galstyan. Namely, we consider two nonempty subsets $A$ and $B$ of a metric space $X$, and construct one-parametric family $F_r$ of subsets obtained by intersection between $B$ and closed $r$-neighborhood of $A$, where $r$ is bigger than the infimum distance between the sets $A$ and $B$. In the case where $B$ is compact, we show that this intersection, considered as a mapping, is right semicontinuously on $r$ in the topology generated by Hausdorff distance. Moreover, if $A$ and $B$ are convex subsets of a normed space $X$, then we prove that $F_r$ depends continuously on $r$ in such topology if and only if the Hausdorff distance between different sets $F_r$ is finite. We also show that for normed spaces $X$ of dimension $2$ or less, the latter condition is automatically fulfilled. For dimension $3$ and hence for bigger ones, we construct an example in which the Hausdorff distance between different $F_r$ is always infinite.

math.MG↗
arXiv2022

Gromov-Hausdorff Distance and Borsuk Number

The aim of this paper is to demonstrate relations between Gromov-Hausdorff distance properties and the Borsuk Conjecture. The Borsuk number of a given bounded metric space $X$ is the infimum of cardinal numbers $n$ such that $X$ can be partitioned into $n$ smaller parts (in the sense of diameter). An exact formula for the Gromov-Hausdorff distance between bounded metric spaces is obtained under the assumptions that the diameter and the cardinality of one space is less than the diameter and the Borsuk number of the other one, respectively. Using Bacon equivalence results between Lusternik-Schnirelmann and Borsuk Problems several corollaries are obtained.

math.GN↗
arXiv2018

Isometry Group of Gromov--Hausdorff Space

The present paper is devoted to investigation of the isometry group of the Gromov-Hausdorff space, i.e., the metric space of compact metric spaces considered up to an isometry and endowed with the Gromov-Hausdorff metric. The main goal is to present a proof of the following theorem by George Lowther (2015): The isometry group of the Gromov-Hausdorff space is trivial. Unfortunately, the author himself has not publish an accurate text for 2 years passed from the publication of draft (that is full of excellent ideas mixed with unproved and wrong statements) in the https://mathoverflow.net/ blog (see the exact reference in he bibliography).

math.MG↗
arXiv2016

Steiner Ratio and Steiner-Gromov Ratio of Gromov-Hausdorff Space

In the present paper we investigate the metric space $\cal M$ consisting of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that for any finite subset $M$ from a sufficiently small neighborhood of a generic finite metric space, providing $M$ consists of finite metric spaces with the same number of points, each Steiner minimal tree in $\cal M$ connecting $M$ is a minimal filling for $M$. As a consequence, we prove that the both Steiner ratio and Gromov-Steiner ratio of $\cal M$ are equal to $1/2$.

math.MG↗
arXiv2016

Local Structure of Gromov-Hausdorff Space, and Isometric Embeddings of Finite Metric Spaces into this Space

We investigate the geometry of the family $\cal M$ of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that sufficiently small neighborhoods of generic finite spaces in the subspace of all finite metric spaces with the same number of points are isometric to some neighborhoods in the space ${\mathbb R}^N_{\infty}$, i.e., in the space ${\mathbb R}^N$ with the norm $\|(x_1,\ldots,x_N)\|=\max_i|x_i|$. As a corollary, we get that each finite metric space can be isometrically embedded into $\cal M$ in such a way that its image belongs to a subspace consisting of all finite metric spaces with the same number $k$ of points. If the initial space has $n$ points, then one can take $k$ as the least possible integer with $n\le k(k-1)/2$.

math.MG↗
arXiv2016

Gromov--Hausdorff Distance, Irreducible Correspondences, Steiner Problem, and Minimal Fillings

We introduce irreducible correspondences that enables us to calculate the Gromov--Hausdorff distances effectively. By means of these correspondences, we show that the set of all metric spaces each consisting of no more than $3$ points is isometric to a polyhedral cone in the space $R^3$ endowed with the maximum norm. We prove that for any $3$-point metric space such that all the triangle inequalities are strict in it, there exists a neighborhood such that the Steiner minimal trees (in Gromov-Hausdorff space) with boundaries from this neighborhood are minimal fillings, i.e., it is impossible to decrease the lengths of these trees by isometrically embedding their boundaries into any other ambient metric space. On the other hand, we construct an example of $3$-point boundary whose points are $3$-point metric spaces such that its Steiner minimal tree in the Gromov-Hausdorff space is not a minimal filling. The latter proves that the Steiner subratio of the Gromov-Hausdorff space is less than 1. The irreducible correspondences enabled us to create a quick algorithm for calculating the Gromov-Hausdorff distance between finite metric spaces. We carried out a numerical experiment and obtained more precise upper estimate on the Steiner subratio: we have shown that it is less than $0.857$.

math.MG↗
arXiv2016

Realizations of Gromov-Hausdorff Distance

It is shown that for any two compact metric spaces there exists an "optimal" correspondence which the Gromov-Hausdorff distance is attained at. Each such correspondence generates isometric embeddings of these spaces into a compact metric space such that the Gromov-Hausdorff distance between the initial spaces is equal to the Hausdorff distance between their images. Also, the optimal correspondences could be used for constructing the shortest curves in the Gromov-Hausdorff space in exactly the same way as it was done by Alexander Ivanov, Nadezhda Nikolaeva, and Alexey Tuzhilin in arXiv:1504.03830, where it is proved that the Gromov-Hausdorff space is geodesic. Notice that all proofs in the present paper are elementary and use no more than the idea of compactness.

math.MG↗
arXiv2016

Fermat-Steiner Problem in the Metric Space of Compact Sets endowed with Hausdorff Distance

The Fermat-Steiner problem consists in finding all points in a metric space $Y$ such that the sum of distances from each of them to the points from some fixed finite subset of $Y$ is minimal. This problem is investigated for the metric space $Y=H(X)$ of compact subsets of a metric space $X$, endowed with the Hausdorff distance. For the case of a proper metric space $X$ a description of all compacts $K\in H(X)$ which the minimum is attained at is obtained. In particular, the Steiner minimal trees for three-element boundaries are described. We also construct an example of a regular triangle in $H(R^2)$, such that all its shortest trees have no "natural" symmetry.

math.MG↗
arXiv2015

Analytic Deformations of Minimal Networks

A behavior of extreme networks under deformations of their boundary sets is investigated. It is shown that analyticity of a deformation of boundary set guarantees preservation of the networks types for minimal spanning trees, minimal fillings and so-called stable shortest trees in the Euclidean space.

math.DG↗
arXiv2015

The Gromov-Hausdorff Metric on the Space of Compact Metric Spaces is Strictly Intrinsic

It is proved that the Gromov-Hausdorff metric on the space of compact metric spaces considered up to an isometry is strictly intrinsic, i.e., the corresponding metric space is geodesic. In other words, each two points of this space (each two compact metric spaces) can be connected by a geodesic. For finite metric spaces a geodesic is constructed explicitly.

math.MG↗
arXiv2014

Branched Coverings and Steiner Ratio

For a branched locally isometric covering of metric spaces with intrinsic metrics, it is proved that the Steiner ratio of the base is not less than the Steiner ratio of the total space of the covering. As applications, it is shown that the Steiner ratio of the surface of an isosceles tetrahedron is equal to the Steiner ratio of the Euclidean plane, and that the Steiner ratio of a flat cone with angle of $2π/k$ at its vertex is also equal to the Steiner ratio of the Euclidean plane.

math.MG↗