Distance Geometry in Quasihypermetric Spaces. I

Details Distance Geometry in Quasihypermetric Spaces. I Distance Geometry in Quasihypermetric Spaces. I

2008-09-04

arxiv.org/abs/0809.0740v1

Mathematics

Metric Geometry

To appear in the Bulletin of the Australian Mathematical Society; 25 pages. References [27] and [28] are arXiv:0809.0744v1 [ma

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. The metric space $(X, d)$ is quasihypermetric if for all $n \in \N$, all $\alpha_1, ..., \alpha_n \in \R$ satisfying $\sum_{i=1}^n \alpha_i = 0$ and all $x_1, ..., x_n \in X$, one has $\sum_{i,j=1}^n \alpha_i \alpha_j d(x_i, x_j) \leq 0$. Without the quasihypermetric property $M(X)$ is infinite, while with the property a natural semi-inner product structure becomes available on $\mathcal{M}_0(X)$, the subspace of $\mathcal{M}(X)$ of all measures of total mass 0. This paper explores: operators and functionals which provide natural links between the metric structure of $(X, d)$, the semi-inner product space structure of $\mathcal{M}_0(X)$ and the Banach space $C(X)$ of continuous real-valued functions on $X$; conditions equivalent to the quasihypermetric property; the topological properties of $\mathcal{M}_0(X)$ with the topology induced by the semi-inner product, and especially the relation of this topology to the weak-$*$ topology and the measure-norm topology on $\mathcal{M}_0(X)$; and the functional-analytic properties of $\mathcal{M}_0(X)$ as a semi-inner product space, including the question of its completeness. A later paper [Peter Nickolas and Reinhard Wolf, Distance Geometry in Quasihypermetric Spaces. II] will apply the work of this paper to a detailed analysis of the constant $M(X)$.

Affiliated with

Also associated with

No associations

Rating

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

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-141433