Quasi-isometric modification of Gromov-Hausdorff distance

TL;DR

This paper introduces a new distance measure for comparing quasi-isometric spaces, explores its properties, and shows that the class of all metric spaces is path-connected under this distance.

Contribution

It defines a Gromov-Hausdorff-like distance for quasi-isometric spaces and proves the path-connectedness of the space of all metric spaces.

Findings

The new distance enables comparison of arbitrary quasi-isometric spaces.

Properties preserved under limits with respect to this distance are investigated.

The class of all metric spaces is path-connected under this distance.

Abstract

We define a distance analogous to the Gromov-Hausdorff distance that enables the comparison of arbitrary quasi-isometric spaces. We also investigate properties preserved under limits with respect to this distance, as well as properties of the entire class of metric spaces equipped with this distance. For this purpose, we introduce the notion of quasi-isometric distortion for correspondences. Using this notion, we prove that the class of all metric spaces is path-connected; in fact, any two metric spaces can be connected by a curve of finite length.