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Alexey A. Tuzhilin

Publications and source records attributed to Alexey A. Tuzhilin.

16 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.

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arXiv2025

Fundamentals of Theory of Continuous Gromov--Hausdorff distance

The Gromov--Hausdorff distance (hereinafter referred to as the GH-distance) is a measure of non-isometricity of metric spaces. In this paper, we study a modification of this distance that also takes topological differences into account. The resulting function of pairs of metric spaces is called the continuous GH-distance. We show that many basic properties of the classical GH-distance also hold in the continuous case. However, the continuous GH-distance, distinguishing between topologies, can differ significantly from the classical one. We will provide numerous examples of this distinction and demonstrate the role of topological dimension here. In particular, we will prove that the continuous GH-distance, like the classical one, is intrinsic, but, unlike the classical one, it is incomplete. Since we are dealing with all metric spaces, we will show, within the framework of the von Neumann-Bernays-Gödel set theory, how topological concepts can be transferred to proper classes.

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arXiv2022

Calculating Gromov-Hausdorff Distance by means of Borsuk Number

The purpose of this article is to demonstrate the connection between the properties of the Gromov--Hausdorff distance 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 assumption that the diameter and cardinality of one space are less than the diameter and Borsuk number of another, respectively. Using the results of Bacon's equivalence between the Lusternik--Schnirelmann and Borsuk problems, several corollaries are obtained.

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arXiv2022

Isometric Embeddings of Bounded Metric Spaces into the Gromov-Hausdorff Class

It is shown that any bounded metric space can be isometrically embedded into the Gromov--Hausdorff metric class GH. This result is a consequence of local geometry description of the class GH in a sufficiently small neighborhood of a generic metric space. This description is interesting in itself. The technique of optimal correspondences and their distortions is used.

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arXiv2021

Gromov-Hausdorff class: its completeness and cloud geometry

The paper is devoted to the study of the Gromov-Hausdorff proper class, consisting of all metric spaces considered up to isometry. In this class, a generalized Gromov-Hausdorff pseudometric is introduced and the geometry of the resulting space is investigated. The first main result is a proof of the completeness of the space, i.e., that all fundamental sequences converge in it. Then we partition the space into maximal proper subclasses consisting of spaces at a finite distance from each other. We call such subclasses clouds. A multiplicative similarity group operates on clouds, multiplying all the distances of each metric space by some positive number. We present examples of similarity mappings transferring some clouds into another ones. We also show that if a cloud contains a space that remains at zero distance from itself under action of all similarities, then such a cloud contracted to this space. In the final part, we investigate subsets of the real line with respect to their behavior under various similarities.

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arXiv2021

Nonparametric estimations and the diffeological Fisher metric

In this paper, first, we survey the concept of diffeological Fisher metric and its naturality, using functorial language of probability morphisms, and slightly extending Lê's theory in \cite{Le2020} to include weakly $C^k$-diffeological statistical models. Then we introduce the resulting notions of the diffeological Fisher distance, the diffeological Hausdorff--Jeffrey measure and explain their role in classical and Bayesian nonparametric estimation problems in statistics.

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arXiv2021

Gromov-Hausdorff Distance Between Segment and Circle

We calculate the Gromov--Hausdorff distance between a line segment and a circle in the Euclidean plane. To do that, we introduced a few new notions like round spaces and nonlinearity degree of a metric space.

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arXiv2020

Lectures on Hausdorff and Gromov-Hausdorff Distance Geometry

The course was given at Peking University, Fall 2019. We discuss the following subjects: (1) Introduction to general topology, hyperspaces, metric and pseudometric spaces, graph theory. (2) Graphs in metric spaces, minimum spanning tree, Steiner minimal tree, Gromov minimal filling. (3) Hausdorff distance, Vietoris topology, Limits theory, inheritance of completeness, total boundedness, compactness by hyperspaces. (4) Gromov-Hausdorff distance, triangle inequality, positive definiteness for isometry classes of compact spaces, counterexample for boundedly compact spaces. (5) Gromov-Hausdorff distance for separable spaces in terms of their isometric images in \ell_\infty, correspondences, Gromov-Hausdorff distance in terms of correspondences. (6) Epsilon-isometries and Gromov-Hausdorff distance. (7) Irreducible correspondences and Gromov-Hausdorff distance. (8) Gromov-Hausdorff convergence, inheritance of metric and topological properties while Gromov-Hausdorff convergence. (9) Gromov-Hausdorff space (GH-space), optimal correspondences, existence of closed optimal correspondences for compact metric spaces, GH-space is geodesic. (10) Cover number, packing number, total boundedness, completeness, and separability of GH-space. (11) mst-spectrum in terms of GH-distances to simplexes, Steiner problem in GH-space. (12) GH-distance to simplexes with more points, GH-distance to simplexes with at most the same number of points. (13) Generalized Borsuk problem, solution of Generalized Borsuk problem in terms of GH-distances, clique covering number and chromatic number of simple graphs, their dualities, calculating these numbers in terms of GH-distances.

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arXiv2019

The Gromov-Hausdorff Distances between Simplexes and Ultrametric Spaces

In the present paper we investigate the Gromov--Hausdorff distances between a bounded metric space $X$ and so called simplex, i.e., a metric space all whose non-zero distances are the same. In the case when the simplex's cardinality does not exceed the cardinality of $X$, a new formula for this distance is obtained. The latter permits to derive an exact formula for the distance between a simplex and an ultrametric space.

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arXiv2017

Hausdorff Measure: Lost in Translation

In the present article we describe how one can define Hausdorff measure allowing empty elements in coverings, and using infinite countable coverings only. In addition, we discuss how the use of different nonequivalent interpretations of the notion "countable set", that is typical for classical and modern mathematics, may lead to contradictions.

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arXiv2016

Who Invented the Gromov-Hausdorff Distance?

One of the most beautiful notions of metric geometry is the Gromov-Hausdorff distance which measures the difference between two metric spaces. To define the distance, let us isometrically embed these spaces into various metric spaces and measure the Hausdorff distance between their images. The best matching corresponds to the least Hausdorff distance. The idea to compare metric spaces in such a way was described in M.Gromov publications dating back to 1981. It was shown that this distance being restricted to isometry classes of compact metric spaces forms a metric which is now called Gromov-Hausdorff metric. However, whether M.Gromov was the first who introduced this metric? It turns out that 6 years before these Gromov's works, in 1975, another mathematician, namely, David Edwards published a paper in which he defined this metric in another way. Also, Edwards found and proved the basic properties of the distance. It seems unfair that almost no one mentions this Edwards's paper including well-known specialists in metric geometry. The main goal of the present publication is to fill this gap.

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arXiv2016

Local Structure of Gromov-Hausdorff Space near Finite Metric Spaces in General Position

We investigate the local structure of the space $\mathcal{M}$ consisting of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We consider finite metric spaces of the same cardinality and suppose that these spaces are in general position, i.e., all nonzero distances in each of the spaces are distinct, and all triangle inequalities are strict. We show that sufficiently small balls in $\mathcal{M}$ centered at these spaces and having the same radii are isometric. As consequences, we prove that the cones over such spaces (with the vertices at one-point space) are isometrical; the isometry group of each sufficiently small ball centered at a general position $n$-points space, $n\ge3$, contains a subgroup isomorphic to the group $S_n$ of permutations of a set containing $n$ points.

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arXiv2016

Geometry of Compact Metric Space in Terms of Gromov-Hausdorff Distances to Regular Simplexes

In the present paper we investigate geometric characteristics of compact metric spaces, which can be described in terms of Gromov-Hausdorff distances to simplexes, i.e., to finite metric spaces such that all their nonzero distances are equal to each other. It turns out that these Gromov-Hausdorff distances depend on some geometrical characteristics of finite partitions of the compact metric spaces; some of the characteristics can be considered as a natural analogue of the lengths of edges of minimum spanning trees. As a consequence, we constructed an unexpected example of a continuum family of pairwise non-isometric finite metric spaces with the same distances to all simplexes.

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arXiv2016

Calculation of Minimum Spanning Tree Edges Lengths using Gromov--Hausdorff Distance

In the present paper we show how one can calculate the lengths of edges of a minimum spanning tree constructed for a finite metric space, in terms of the Gromov-Hausdorff distances from this space to simplices of sufficiently large diameter. Here by simplices we mean finite metric spaces all of whose nonzero distances are the same. As an application, we reduce the problems of finding a Steiner minimal tree length or a minimal filling length to maximization of the total distance to some finite number of simplices considered as points of the Gromov-Hausdorff space.

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arXiv2011

One-dimensional minimal fillings with negative edge weights

Ivanov and Tuzhilin started an investigation of a particular case of Gromov Minimal Fillings problem (generalized to the case of stratified manifolds). Weighted graphs with non-negative weight function were used as minimal fillings of finite metric spaces. In the present paper we introduce generalized minimal fillings, i.e. minimal fillings where the weight function is not necessarily non-negative. We prove that for any finite metric space its minimal filling has the minimum weight in the class of all generalized fillings of the space.

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