Distance geometry in quasihypermetric spaces. II

✍ Scribed by Peter Nickolas; Reinhard Wolf

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 135 KB
• Volume: 284
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200710206

✦ Synopsis

Abstract

Let (X, d) be a compact metric space and let ${\cal M}(X)$ denote the space of all finite signed Borel measures on X. Define \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$I :{\cal M}(X)\rightarrow {\mathbb R}$\end{document} by I(μ) = ∫~X~∫~X~d(x, y) __d__μ(x)__d__μ(y), and set $M(X) = \sup I(\mu)$, where μ ranges over the collection of signed measures in ${\cal M}(X)$ 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 ${\cal M}(X)$ 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). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim