Continuity of Magnitude at Skew Finite Subsets of ${\ell }_1^N$

Abstract

Magnitude is an isometric invariant of metric spaces introduced by Leinster. Although magnitude is nowhere continuous on the Gromov-Hausdorff space of finite metric spaces, continuity results are possible if we restrict the ambient space. In this paper, we focus on ${\ell }_1^N$ and prove that magnitude is continuous at every skew finite subset of ${\ell }_1^N$, that is, at every finite set whose coordinate projections are injective. For such sets, we analyze cubical thickenings and derive an explicit formula for their weight measures. This yields a formula for the magnitude of these thickenings, which we use to prove that their magnitude converges to that of the underlying finite set. Since skew finite subsets of ${\ell }_1^N$ form an open and dense subset of the space of all finite subsets, magnitude is continuous on an open dense subset of the space of finite subsets of ${\ell }_1^N$.