Related papers: Ultrametric spaces and clouds
Considering the Teichm\"uller space of a surface equipped with Thurston's Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point…
For a metric space $X$ we study metrics on the two copies of $X$. We define composition of such metrics and show that the equivalence classes of metrics are a semigroup $M(X)$ Our main result is that $M(X)$ is an inverse semigroup,…
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the…
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.…
In this paper we prove an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality. These include all Banach spaces and all complete, simply-connected metric spaces of non-positive curvature in…
In the context of metric geometry, we introduce a new necessary and sufficient condition for the convergence of an inductive sequence of quantum compact metric spaces for the Gromov-Hausdorff propinquity, which is a noncommutative analogue…
In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group $\Iso(\U)$ of isometries of…
Reflection in a line in Euclidean 3-space defines an almost paracomplex structure on the space of all oriented lines, isometric with respect to the canonical neutral Kaehler metric. Beyond Euclidean 3-space, the space of oriented geodesics…
We prove that if a sequence of geodesically complete CAT$(0)$-spaces $X_j$ with uniformly cocompact discrete groups of isometries converges in the Gromov-Hausdorff sense to $X_\infty$, then the dimension of the maximal Euclidean factor…
We show that uniform lattices of isometries of products of real hyperbolic spaces act properly discontinuously and cocompactly on a median space. For lattices in products of at least two factors, this is the strongest degree of…
The Heisenberg group $\mathbb{H}$ equipped with a sub-Riemannian metric is one of the most well known examples of a doubling metric space which does not admit a bi-Lipschitz embedding into any Euclidean space. In this paper we investigate…
We consider the notion of Borel reducibility between pseudometrics on standard Borel spaces introduced and studied recently by C\'{u}th, Doucha and Kurka, as well as the notion of an orbit pseudometric, a continuous version of the notion of…
Prompted by a recent question of G. Hjorth as to whether a bounded Urysohn space is indivisible, that is to say has the property that any partition into finitely many pieces has one piece which contains an isometric copy of the space, we…
The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any $0<\beta<\alpha$, any compact metric space $X$ of Hausdorff dimension $\alpha$…
We study a new bi-Lipschitz invariant \lambda(M) of a metric space M; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are enlarged by a factor…
Let X be quasi-isometric to either the mapping class group equipped with the word metric, or to Teichmuller space equipped with either the Teichmuller metric or the Weil-Petersson metric. We introduce a unified approach to study the coarse…
The notion of the ultrametrics can be considered as a zero-dimensional analogue of ordinary metrics, and it is expected to prove ultrametric versions of theorems on metric spaces. In this paper, we provide ultrametric versions of the…
We study the metric and topological properties of the space $\mathscr{D}(G)$ of left-invariant hyperbolic pseudometrics on the non-elementary hyperbolic group $G$ that are quasi-isometric to a word metric, up to rough similarity. This space…
We show that all the standard distances from metric geometry and functional analysis, such as Gromov-Hausdorff distance, Banach-Mazur distance, Kadets distance, Lipschitz distance, Net distance, and Hausdorff-Lipschitz distance have all the…
An $e^\epsilon$-Lipschitz and co-Lipschitz map, as a metric analogue of an $\epsilon$-Riemannian submersion, naturally arises from a sequence of Alexandrov spaces with curvature uniformly bounded below that converges to a space of only weak…