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PreprintarXiv licencearXiv

Continuity of Magnitude at Finite Subsets of ℓ1N

  • Sara Kališnik
  • Davorin Lešnik
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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 ℓ1N. 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.

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Cite

BibTeXRIS
BibTeX
@misc{kalisnik2026continuity,
  title = {{Continuity of Magnitude at Finite Subsets of $\ell_1^N$}},
  author = {Sara Kališnik and Davorin Lešnik},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26560},
  eprint = {2609.26560},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26560},
}
RIS
TY  - UNPB
TI  - Continuity of Magnitude at Finite Subsets of ℓ_1^N
AU  - Sara Kališnik
AU  - Davorin Lešnik
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26560
UR  - https://arxiv.org/abs/2609.26560
AB  - 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 ℓ_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.
ER  -