This is an announcement for the paper "Positive definite metric spaces" by Mark W. Meckes.

Abstract: Magnitude is a numerical invariant of finite metric spaces, recently introduced by T.\ Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of topological spaces. It has been extended to infinite metric spaces in several a priori distinct ways. This paper develops the theory of a class of metric spaces, positive definite metric spaces, for which magnitude is more tractable than in general. In particular, it is shown that all the proposed definitions of magnitude coincide for compact positive definite metric spaces. Some additional results are proved about the behavior of magnitude as a function of such spaces, and a number of examples of positive definite metric spaces are found, including all subsets of $L_p$ for $p\le 2$ and Euclidean spheres equipped with the geodesic distance. Finally, some facts about the magnitude of compact subsets of $\ell_p^n$ for $p \le 2$ are proved, generalizing results of Leinster for $p=1,2$, using properties of these spaces which are somewhat stronger than positive definiteness.

Archive classification: math.MG math.FA math.GN

Submitted from: mark.meckes@case.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1012.5863

or

http://arXiv.org/abs/1012.5863