Convergence in the space of compact labeled metric spaces

2022/10/02 by Reijo Jaakkola, Antti Kykkänen, Jaakkola, Reijo +1
Mathematics · #51F99 #53C23 #54E45 #Analytic and geometric function theory #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2210.00535

openalex publication_date 2022/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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