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

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.

14 pages, 4 figures. Version 2: includes acknowledgements and an updated reference, and a few typos are corrected. Version 3: references updated; minor corrections and improvements following referee suggestions. This is the final version, to appear in the Bulletin of the London Mathematical Society