Simon Willerton and I have a new preprint out: "On the asymptotic magnitude of subsets of Euclidean space" http://arxiv.org/abs/0908.1582 Abstract: 'Magnitude' is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets. (Some of you have heard me talk about this invariant under the name of "cardinality" rather than "magnitude".) We'd be grateful for any comments. There's a discussion going at http://golem.ph.utexas.edu/category/2009/08/asymptotics_of_the_magnitude_o.h... Best wishes, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]