# A Unified Framework for Generalizing the Gromov-Hausdorff Metric

## TL;DR

This paper introduces a unified framework for generalizing the Gromov-Hausdorff metric to include additional structures on metric spaces, enabling comprehensive analysis of complex random metric spaces.

## Contribution

It develops a general abstract framework that unifies various Gromov-Hausdorff-type metrics and studies their topological properties for different classes of metric spaces.

## Key findings

- Unified framework for Gromov-Hausdorff metrics with additional structures
- Proved completeness and separability under certain conditions
- Includes new examples and applications to random metric spaces

## Abstract

In this paper, an approach for generalizing the Gromov-Hausdorff metric is presented, which applies to metric spaces equipped with some additional structure. A special case is the Gromov-Hausdorff-Prokhorov metric between measured metric spaces. This abstract framework unifies several existing Gromov-Hausdorff-type metrics for metric spaces equipped with a measure, a point, a closed subset, a curve, a tuple of such structures, etc. Along with reviewing these special cases in the literature, several new examples are also presented. Two frameworks are provided, one for compact metric spaces and the other for boundedly-compact pointed metric spaces. In both cases, a Gromov-Hausdorff-type metric is defined and its topological properties are studied. In particular, completeness and separability is proved under some conditions. This enables one to study random metric spaces equipped with additional structures, which is the main motivation of this work.