Distance Geometry in Quasihypermetric Spaces. II

Abstract

Let $(X,d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I:\mathcal{M}(X)\to \mathbb{R}$ by \[ I(μ) = \int_X \int_X d(x,y) dμ(x) dμ(y), \] and set $M(X)=\sup I(\mu )$, where $\mu$ ranges over the collection of signed measures in $\mathcal{M}(X)$ of total mass 1. This paper, with an earlier and a subsequent paper [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and III], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $\mathcal{M}(X)$ when equipped with a natural semi-inner product. Using the work of the earlier paper, this paper explores measures which attain the supremum defining $M(X)$, sequences of measures which approximate the supremum when the supremum is not attained and conditions implying or equivalent to the finiteness of $M(X)$.

15 pages. References [8] and [9] are arXiv:0809.0740v1 [math.MG] and arXiv:0809.0746v1 [math.MG]