Distance Geometry in Quasihypermetric Spaces. III

Details Distance Geometry in Quasihypermetric Spaces. III Distance Geometry in Quasihypermetric Spaces. III

2008-09-04

arxiv.org/abs/0809.0746v1

Mathematics

Metric Geometry

20 pages. References [10] and [11] are arXiv:0809.0740v1 [math.MG] and arXiv:0809.0744v1 [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 two earlier papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and II], 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. Specifically, this paper explores links between the properties of $M(X)$ and metric embeddings of $X$, and the properties of $M(X)$ when $X$ is a finite metric space.

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