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arXiv · 2603.04271

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

Sara Kalisnik·Davorin Lesnik
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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 $\ell_1^N$.

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Sara Kalisnik, Davorin Lesnik. 2026-03-04. Continuity of Magnitude at Skew Finite Subsets of $\ell_1^N$. https://arxiv.org/abs/2603.04271

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