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

Understand

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.