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arXiv · 2312.14497

The small-scale limit of magnitude and the one-point property

Emily Roff·Masahiko Yoshinaga
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Abstract

The magnitude of a metric space is a real-valued function whose parameter controls the scale of the metric. A metric space is said to have the one-point property if its magnitude converges to 1 as the space is scaled down to a point. Not every finite metric space has the one-point property: to date, exactly one example has been found of a finite space for which the property fails. Understanding the failure of the one-point property is of interest in clarifying the interpretation of magnitude and its stability with respect to the Gromov--Hausdorff topology. We prove that the one-point property holds generically for finite metric spaces, but that when it fails, the failure can be arbitrarily bad: the small-scale limit of magnitude can take arbitrary real values greater than 1.

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Emily Roff, Masahiko Yoshinaga. 2023-12-22. The small-scale limit of magnitude and the one-point property. https://arxiv.org/abs/2312.14497

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