Convergence of Magnitude of Finite Positive Definite Metric Spaces

Abstract: The magnitude of metric spaces does not to 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. The main theorem of this article specifies the part of the partition for which, for a convergent sequence of finite metric spaces lying in the part, the magnitude converges to that of the limit. This study takes advantage of the geometric interpretation of magnitude as the circumradius of the corresponding finite Euclidean subset, called similarity embedding, as recently proposed by other studies. Such a transformation is especially useful for constructing counterexamples as we can depend on Euclidean geometric intuition.