Tractable Metric Spaces and Magnitude Continuity

TL;DR

This paper introduces tractable metric spaces, characterizes them, and proves continuity and Lipschitz properties of magnitude within these spaces, enhancing understanding of magnitude's stability.

Contribution

It defines tractable metric spaces, provides their characterization, and proves new continuity and Lipschitz results for magnitude in these spaces.

Findings

Magnitude is continuous on compact subsets of R under the Hausdorff metric.

Magnitude function is Lipschitz on bounded subspaces of R.

New proof of magnitude continuity on certain metric spaces.

Abstract

Magnitude is an isometric invariant of metric spaces introduced by Leinster. Since its inception, it has inspired active research into its connections with integral geometry, geometric measure theory, fractal dimensions, persistent homology, and applications in machine learning. In particular, when it comes to applications, continuity and stability of invariants play an important role. Although it has been shown that magnitude is nowhere continuous on the Gromov--Hausdorff space of finite metric spaces, positive results are possible if we restrict the ambient space. In this paper, we introduce the notion of tractable metric spaces, provide a characterization of these spaces, and establish several continuity results for magnitude in this setting. As a consequence, we offer a new proof of a known result stating that magnitude is continuous on the space of compact subsets of $\mathbb{R}$ with respect to the Hausdorff metric. Furthermore, we show that the magnitude function is Lipschitz when restricted to bounded subspaces of $\mathbb{R}$.