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

Sara Kalisnik, Davorin Lesnik

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 $\ell_1^N$.

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 and prove that magnitude is continuous at every skew finite subset of , 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 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 .
Paper Structure (11 sections, 11 theorems, 61 equations)

This paper contains 11 sections, 11 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. The Magnitude of Finite Metric Spaces
  5. Positive Definite Metric Spaces
  6. The Magnitude of Compact Metric Spaces
  7. Magnitude Continuity via Thickenings
  8. Cubes and Skew Finite Subsets of $\ell_1^N$
  9. Weight Measure of a Union of Cubes
  10. Continuity of Magnitude at Skew Finite Subsets
  11. Conclusion

Key Result

Lemma 3.2

Let $M$ be a tractable metric space and $K \in \mathrm{Cmp}_{+}(M)$. The following statements are equivalent.

Theorems & Definitions (45)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 35 more