Heuristic and computer calculations for the magnitude of metric spaces

Abstract

The notion of the magnitude of a compact metric space was considered in arXiv:0908.1582 with Tom Leinster, where the magnitude was calculated for line segments, circles and Cantor sets. In this paper more evidence is presented for a conjectured relationship with a geometric measure theoretic valuation. Firstly, a heuristic is given for deriving this valuation by considering 'large' subspaces of Euclidean space and, secondly, numerical approximations to the magnitude are calculated for squares, disks, cubes, annuli, tori and Sierpinski gaskets. The valuation is seen to be very close to the magnitude for the convex spaces considered and is seen to be 'asymptotically' close for some other spaces.