Distance geometry in quasihypermetric spaces. III

✍ Scribed by Peter Nickolas; Reinhard Wolf

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 157 KB
• Volume: 284
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200810216

✦ Synopsis

Abstract

Let (X, d) be a compact metric space and let \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathcal {M}(X)$\end{document} denote the space of all finite signed Borel measures on X. Define \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$I :\mathcal {M}(X)\longrightarrow \mathbb {R}$\end{document} by

and set $M(X) = \sup I(\mu ),$ where μ ranges over the collection of signed measures in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathcal {M}(X)$\end{document} 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 \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathcal {M}(X)$\end{document} 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. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim