Continuity of Magnitude at Skew Finite Subsets of $\ell_1^N$

TL;DR

This paper proves that the magnitude, an isometric invariant, is continuous at skew finite subsets of ^N, and provides explicit formulas for their magnitude and convergence properties.

Contribution

It establishes the continuity of magnitude at skew finite subsets of ^N and derives explicit formulas for their magnitude and weight measures.

Findings

Magnitude is continuous at skew finite subsets of ^N.

Explicit formulas for the magnitude of cubical thickenings.

Magnitude converges to that of the underlying finite set.

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 $\ell_1^N$.