Convergence of magnitude of finite positive definite metric spaces

TL;DR

This paper investigates the conditions under which the magnitude of finite positive definite metric spaces is continuous with respect to the Gromov-Hausdorff distance, using geometric interpretations and neighborhood restrictions.

Contribution

It provides a new criterion based on cardinality for the continuity of magnitude in finite metric spaces, utilizing Euclidean geometric insights.

Findings

Identifies a cardinality-based condition for magnitude continuity.

Uses geometric interpretation as circumradius to analyze metric spaces.

Provides counterexamples based on Euclidean geometry.

Abstract

The magnitude of metric spaces does not appear to possess a simple, convenient continuity property, and previous studies have presented affirmative results under additional constraints or weaker notions, as well as counterexamples. In this vein, we discuss the continuity of magnitude of finite positive definite metric spaces with respect to the Gromov-Hausdorff distance, but with a restriction of the domain based on a canonical partition of a sufficiently small neighborhood of a finite metric space. As a result, the main theorem of this article explains a condition on the cardinality of metric spaces that determines the continuity of magnitude. This study takes advantage of the geometric interpretation of magnitude as the circumradius of the corresponding finite Euclidean subset. Such a transformation is especially useful for constructing counterexamples, as we can depend on Euclidean geometric intuition.