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.
The present article addresses to everyone who starts working with (pointed) Gromov-Hausdorff convergence. In the major part, both Gromov-Hausdorff convergence of compact and of pointed metric spaces are introduced and investigated. Moreover, the relation of sublimits occurring with pointed Gromov-Hausdorff convergence and ultralimits is discussed.
We have constructed a realization of rectilinear geodesic (in the sense of~\cite{Memoli2018}), lying in Gromov-Hausdorff space, as a shortest geodesic w.r.t. the Hausdorff distance in an ambient metric space.
In this paper we prove that generic metric spaces are everywhere dense in the proper class of all metric spaces endowed with the Gromov-Hausdorff distance.
In this paper, we introduce a pseudometric on the family of isometry classes of (extended) metric spaces. Using it, we obtain a natural compactification of the Gromov-Hausdorff space, which is compatible with ultralimit.
It is shown that each finite family of finite metric spaces, being considered as a subset of Gromov--Hausdorff space, can be connected by a Steiner minimal tree.
In this paper we introduce a notion of the Gromov-Hausdorff distance with boundary, denoted by $d_{GHB}$, to construct a framework of convergence of noncomplete metric spaces. We show that a class of bounded $A$-uniform spaces with diameter bounded from below is a complete metric space with respect to $d_{GHB}$. As an application we show the stability of Gromov hyperbolicity, roughly starlike property, uniformization, quasihyperbolization, and boundary of Gromov hyperbolic spaces under appropriate notions of convergence and assumptions.
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.
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:1504.038…
The paper is devoted to geometrical investigation of the Gromov-Hausdorff distance on the classes of all metric spaces and of all bounded metric spaces. The main attention is paid to pass connectivity questions. The pass connected components of the Gromov-Hausdorff class of all metric spaces are described, and the pass connectivity of spheres is proved in several particular cases.
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 $1/2$.
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 (…
We establish universality and ultra-homogeneity of $(\mathcal{U},u_\mathrm{GH})$, the collection of all compact ultrametric spaces endowed with the so-called Gromov-Hausdorff ultrametric. This result also gives rise to a novel construction of the so-called $R$-Uryoshn universal ultrametric space for each countable subset $R\subset\mathbb{R}_{\geq 0}$ containing $0$.
We prove that any measured Gromov-Hausdorff precompact set of metric measure spaces which is contained in a certain set, called a pyramid, is bounded by some metric measure space with respect to the Lipschitz order inside the pyramid. This is proved as a step towards a possible extension of the statement of Gromov, for which we gave a detailed proof in our previous work. Several related results are also obtained.
We provide an alternative, constructive proof that the collection $\mathcal{M}$ of isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance is a geodesic space. The core of our proof is a construction of explicit geodesics on $\mathcal{M}$. We also provide several interesting examples of geodesics on $\mathcal{M}$, including a geodesic between $\mathbb{S}^0$ and $\mathbb{S}^n$ for any $n\geq 1$.
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.
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.
In this paper, we first show that for all four non-negative real numbers, there exists a Cantor ultrametric space whose Hausdorff dimension, packing dimension, upper box dimension, and Assouad dimension are equal to given four numbers, respectively. Next, by constructing topological embeddings of an arbitrary compact metrizable space into the Gromov--Hausdorff space using a direct sum of metrics spaces, we prove that the set of all compact metric spaces possessing prescribed topological dimension, and four dimensions explained above, and the set of all compact ultrametric spaces are path-conne…
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.
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.
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 betw…
Using the wedge sum of metric spaces, for all compact metrizable spaces, we construct a topological embedding of the compact metrizable space into the set of all metric trees in the Gromov--Hausdorff space with finite prescribed values. As its application, we show that the set of all metric trees is path-connected and its all non-empty open subsets have infinite topological dimension.
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…
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…
We first prove that for all compact metrizable spaces, there exists a topological embedding of the compact metrizable space into each of the sets of compact metric spaces which are connected, path-connected, geodesic, or CAT(0), in the Gromov--Hausdorff space with finite prescribed values. As its application, we show that the sets prescribed above are path-connected and their non-empty open subsets have infinite topological dimension. By the same method, we also prove that the set of all proper CAT(0) spaces is path-connected and its non-empty open subsets have infinite topological dimension w…
In this work, a metric is presented on the set of boundedly-compact pointed metric spaces that generates the Gromov-Hausdorff topology. A similar metric is defined for measured metric spaces that generates the Gromov-Hausdorff-Prokhorov topology. This extends previous works which consider only length spaces or discrete metric spaces. Completeness and separability are also proved for these metrics. Hence, they provide the measure theoretic requirements to study random (measured) boundedly-compact pointed metric spaces, which is the main motivation of this work. In addition, we present a general…
We consider the proper class of all metric spaces endowed with the Gromov--Hausdorff distance. Its maximal subclasses, consisting of the spaces on finite distance from each other, we call clouds. Multiplying all distances in a metric space by the same positive real number, we obtain a similarity transformation of the Gromov--Hausdorff class. In our previous work, we observed that with such a transformation, some clouds can jump to others. To characterize the phenomenon, we studied the stabilizers of the similarity action. In this paper, we prove that every cloud with a nontrivial stabilizer ha…
In this paper we study the space $\mathcal{M}$ of all nonempty compact metric spaces considered up to isometry, equipped with the Gromov--Hausdorff distance. We show that each ball in $\mathcal{M}$ with center at the one-point space is convex in the weak sense, i.e., every two points of such a ball can be joined by a shortest curve that belongs to this ball, however, such a ball is not convex in the strong sense: it is not true that every shortest curve joining the points of the ball belongs to this ball. We also show that a ball of sufficiently small radius with center at a space of general p…
The collection $\mathcal{M}$ of all isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance $d_\mathcal{GH}$ is known to be a geodesic space. However, there is no known structural characterization of geodesics in $\mathcal{M}$. In this paper we provide two such characterizations. We first prove that every Gromov-Hausdorff geodesic is in fact a geodesic in the Hausdorff hyperspace of some compact metric space, which we call a Hausdorff geodesic. Inspired by this characterization, we further elucidate a structural connection between Hausdorff geodesics and Wasserstei…
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…
We show continuity under equivariant Gromov-Hausdorff convergence of the critical exponent of discrete, non-elementary, torsion-free, quasiconvex-cocompact groups with uniformly bounded codiameter acting on uniformly Gromov-hyperbolic metric spaces.
We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.
We show that there exists a natural counterpart of the Gromov-Hausdorff metric in the class of ultrametric spaces. It is proved, in particular, that the space of all ultrametric spaces whose metric take values in a fixed countable set is homeomorphic to the space of irrationals.
