Distance Geometry in Quasihypermetric Spaces. II

Details Distance Geometry in Quasihypermetric Spaces. II Distance Geometry in Quasihypermetric Spaces. II

2008-09-04

arxiv.org/abs/0809.0744v1

Mathematics

Metric Geometry

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

Scientific paper

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 $I \colon \mathcal{M}(X) \to \R$ by \[ I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y), \] 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 $M(X)$.

Affiliated with

Also associated with

No associations

Rating

Distance Geometry in Quasihypermetric Spaces. II does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Distance Geometry in Quasihypermetric Spaces. II, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Distance Geometry in Quasihypermetric Spaces. II will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-141456