Related papers: Gromov-Hausdorff Distance for Directed Spaces

We construct an isometric embedding of a bounded set in a Euclidean space into the Gromov-Hausdorff space. In particular, we can embed a bounded and connected Riemannian manifold into the Gromov-Hausdorff space by a bilipschitz map.

In this note, we prove optimal extension results for roughly isometric relations between metric (R-)trees and injective metric spaces. This yields sharp stability estimates, in terms of the Gromov-Hausdorff (GH) distance, for certain metric…

Smocked spaces are a class of metric spaces which were introduced to generalize pulled thread spaces. We investigate convergence of these spaces, showing that if the underlying smocking sets converge in Hausdorff distance and satisfy local…

Inspired by the Kantorovich formulation of optimal transport distance between probability measures on a metric space, Gromov-Wasserstein (GW) distances comprise a family of metrics on the space of isomorphism classes of metric measure…

In this paper we prove that the Gromov--Hausdorff distance between $\mathbb{R}^n$ and its subset $A$ is finite if and only if $A$ is an $\varepsilon$-net in $\mathbb{R}^n$ for some $\varepsilon>0$. For infinite-dimensional Euclidean spaces…

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…

For a convergent series with positive terms, we prove that the $\ell^\infty$ product space of bounded subspaces of the Gromov-Hausdorff space can be isometrically embedded into the Gromov-Hausdorff space, where each subspace consists of…

We present theoretical properties of the space of metric pairs equipped with the Gromov--Hausdorff distance. First, we establish the classical metric separability and the geometric geodesicity of this space. Second, we prove an…

We show that Rieffel's quantum Gromov-Hausdorff distance between two compact quantum metric spaces is not equivalent to the ordinary Gromov-Hausdorff distance applied to the associated state spaces.

Let $G$ be a finite, connected metric graph and let $X\subseteq G$ be a subset. If $X$ is sufficiently dense in $G$, we show that the Gromov--Hausdorff distance matches the Hausdorff distance, namely $d_\gh(G,X)=d_\h(G,X)$. When the metric…

This writeup describes ongoing work on designing and testing a certain family of correspondences between compact metric spaces that we call \emph{embedding-projection correspondences} (EPCs). Of particular interest are EPCs between spheres…

The definition of weighted distance measure involves weights. The paper proposes a weighted distance measure without the help of weights. Here, weights are intrinsically added to the measure, and for this, the concept of metric space is…

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.…

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…

In this paper, we discuss the embeddability of subspaces of the Gromov-Hausdorff space, which consists of isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance, into Hilbert spaces. These embeddings are…

We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the…

The Hausdorff distance is a metric commonly used to compute the set similarity of geometric sets. For sets containing a total of $n$ points, the exact distance can be computed na\"{i}vely in $O(n^2)$ time. In this paper, we show how to…

Comparing two geometric graphs embedded in space is important in the field of transportation network analysis. Given street maps of the same city collected from different sources, researchers often need to know how and where they differ.…

In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with…

In ``Characterization, stability and convergence of hierarchical clustering methods'' by G. E. Carlsson, F. Memoli, the natural way to construct an ultrametric space from a given metric space was presented. It was shown that the…