Year: 2026
Type: preprint
Abstract: Magnitude is an isometric invariant of metric spaces introduced by Leinster in 2011. It is nowhere continuous on the Gromov--Hausdorff space of finite metric spaces, however, positive continuity results do exist if we restrict the ambient space. In this paper, we prove that magnitude is continuous at every finite subset $F$ of $\ell_1^N$. We do this by first deriving the weight measure for a finite union of cubes, i.e.\ cubical thickenings of the points in $F$, and then showing that these thickenings converge to the magnitude of the underlying finite set.
Source: arXiv (Cornell University)
Authors: Kališnik, Sara, Lešnik, Davorin
Institutions: Pennsylvania State University, University of Ljubljana
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214123489