Continuity of magnitude for positive definite metric spaces

Determine whether, for any positive definite metric space W (i.e., every finite subspace has positive-definite similarity matrix), the magnitude function is continuous with respect to the Hausdorff topology on the set of compact subspaces of W.

Background

Building on Meckes’s framework, positive definite metric spaces are a natural class where magnitude exhibits favorable properties (e.g., lower semicontinuity on compact positive definite spaces). The authors highlight Meckes’s question as an avenue toward a stability theorem for magnitude.

A positive resolution would imply continuity for finite subsets of classical ambient spaces such as (ℝ^n, ℓ^p) for 1 ≤ p ≤ 2, which are positive definite, and would directly impact practical applications in data analysis where magnitudes are computed for point clouds embedded in Euclidean or taxicab spaces.