Heuristic and Computer Calculations for the Magnitude of Metric Spaces

This paper is concerned with an empirical look at the magnitude of subspaces of Euclidean space. Magnitude was introduced under the name ‘cardinality’ by Leinster in [4] as a partially defined invariant of finite metric spaces; his motivation came from enriched category theory, but the invariant had already been considered in the biological diversity literature [6]. In [5] we started to extend the definition to compact, non-finite metric spaces by approximating these with finite metric spaces; we considered, in particular, the magnitude of certain subsets of Euclidean space, such as line segments and circles and we showed that in these cases, when the metric space is scaled up in size, the magnitude asymptotically satisfies the inclusion-exclusion principle. This leads to some natural conjectures and the current paper provides numerical and heuristic evidence for these conjectures. The current paper, however, can be read independently of [5]. As hinted by the title, there are two main ingredients to this paper. The first ingredient is a heuristic argument to calculate the contribution to the magnitude from the ‘bulk’ of points in the closure of a ‘large’ open subset X of Euclidean space Rn; this contribution is shown to be roughly vol(X)/n! ωn where ωn is the volume of the unit n-ball. From here one is led to consider, for naturality reasons, the so-called penguin valuation, defined on certain compact subsets of Rn by