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.