Convergence in the space of compact labeled metric spaces

TL;DR

This paper investigates the properties of a labeled Gromov-Hausdorff distance, establishing convergence, completeness, and precompactness results for labeled metric spaces, with applications to travel time inverse problems.

Contribution

It introduces and analyzes the labeled Gromov-Hausdorff distance, extending classical concepts to include boundary points, and provides foundational results for convergence and compactness in this space.

Findings

Proves completeness of the space of labeled metric spaces

Provides characterizations of precompact subsets

Applies results to travel time inverse problems

Abstract

A labeled metric space is intuitively speaking a metric space together with a special set of points to be understood as the geometric boundary of the space. We study basic properties of a recently introduced labeled Gromov-Hausdorff distance, an extension of the classical Gromov-Hausdorff distance, which measures how close two labeled metric spaces are. We provide a toolbox of results characterizing convergence in the labeled Gromov-Hausdorff distance. We obtain a completeness result for the space of labeled metric spaces and precompactness characterizations for subsets of the space. The results are applied to travel time inverse problems in a labeled metric space context.