相关论文: Convex hyperspaces of probability measures and ext…

It is well-known that a paracompact space X is of covering dimension n if and only if any map f from X to a simplicial complex K can be pushed into its n-skeleton. We use the same idea to define dimension in the coarse category. It turns…

We develop some basic Lipschitz homotopy technique and apply it to manifolds with finite asymptotic dimension. In particular we show that the Higson compactification of a uniformly contractible manifold is mod $p$ acyclic in the finite…

A nonnegative number d_infinity, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and…

The aim of this paper is to introduce an asymptotic counterpart of the extension dimension defined by Dranishnikov. The main result establishes a relation between the asymptotic extensional dimension of a proper metric space and extension…

Dranishnikov and Zarichnyi constructed a universal space in the coarse category of spaces of bounded geometry of asymptotic dimension $0$. In this paper we construct universal spaces in the coarse category of separable (respectively,…

In this paper, we generalize Dranishnikov's asymptotic inductive dimension to the setting of coarse proximity spaces. We show that in this more general context, the asymptotic inductive dimension of a coarse proximity space is bigger or…

We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric…

The notion of asymptotic space for an unbounded metric space has been introduced by Micha Gromov in 1980s. It is intended to capture the structure of a metric space at infinity. The most comprehensive definition of asymptotic space is given…

The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and…

We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform…

This paper is devoted to dualization of dimension-theoretical results from the small scale to the large scale. So far there are two approaches for such dualization: one consisting of creating analogs of small scale concepts and the other…

We give an example of a visual Gromov Hyperbolic metric space X with asdimX=2 and dim(bdry X)=0

We prove that any proper, geodesic metric space whose Dehn function grows asymptotically like the Euclidean one has asymptotic cones which are non-positively curved in the sense of Alexandrov, thus are ${\rm CAT}(0)$. This is new already in…

We introduce the notion of large scale inductive dimension for asymptotic resemblance spaces. We prove that the large scale inductive dimension and the asymptotic dimensiongrad are equal in the class of r-convex metric spaces. This class…

The asymptotic dimension of metric spaces is an important notion in geometric group theory introduced by Gromov. The metric spaces considered in this paper are the ones whose underlying spaces are the vertex-sets of graphs and whose metrics…

We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however,…

We introduce a quasi-symmetry invariant of a metric space Z called the capacity dimension. Our main result says that for a visual Gromov hyperbolic space X the asymptotic dimension of X is at most the capacity dimension of its boundary at…

The Gromov-Hausdorff distance is a dissimilarity metric capturing how far two spaces are from being isometric. The Gromov-Prokhorov distance is a similar notion for metric measure spaces. In this paper, we study the topological dimension of…

A complete Riemannian manifold without conjugate points is called asymptotically harmonic if the mean curvature of its horospheres is a universal constant. Examples of asymptotically harmonic manifolds include flat spaces and rank one…

We prove the existence of almost isoperimetric extremisers for two classes of probability measures defined respectively on the Grushin space and a stratified Lie group. It turns out such extremisers can be regarded as a type of anisotropic…