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DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. I

Part of: Other generalizations Distance geometry

Published online by Cambridge University Press:  19 June 2009

PETER NICKOLAS  and
REINHARD WOLF
PETER NICKOLAS*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia (email: peter_nickolas@uow.edu.au)
REINHARD WOLF
Affiliation:
Institut für Mathematik, Universität Salzburg, Hellbrunnerstrasse 34, A-5020 Salzburg, Austria (email: Reinhard.Wolf@sbg.ac.at)
*
✉For correspondence; e-mail: peter˙nickolas@uow.edu.au
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Abstract

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Let (X,d) be a compact metric space and let ℳ(X) denote the space of all finite signed Borel measures on X. Define I:ℳ(X)→ℝ by and set M(X)=sup I(μ), where μ ranges over the collection of signed measures in ℳ(X) of total mass 1. The metric space (X,d) is quasihypermetric if for all n∈ℕ, all α1,…,αn∈ℝ satisfying ∑ i=1nαi=0 and all x1,…,xn∈X, the inequality ∑ i,j=1nαiαjd(xi,xj)≤0 holds. Without the quasihypermetric property M(X) is infinite, while with the property a natural semi-inner product structure becomes available on ℳ0(X), the subspace of 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 ℳ0(X) and the Banach space C(X) of continuous real-valued functions on X; conditions equivalent to the quasihypermetric property; the topological properties of ℳ0(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 ℳ0(X); and the functional-analytic properties of ℳ0(X) as a semi-inner product space, including the question of its completeness. A later paper [P. Nickolas and R. Wolf, Distance geometry in quasihypermetric spaces. II, Math. Nachr., accepted] will apply the work of this paper to a detailed analysis of the constant M(X).

Keywords

compact metric spacefinite metric spacequasihypermetric spacemetric embeddingsigned measuresigned measure of mass zerospaces of measuresdistance geometrygeometric constant

MSC classification

Secondary: 51K05: General theory 31C45: Other generalizations (nonlinear potential theory, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 1 , August 2009 , pp. 1 - 25
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

The authors are grateful for the financial support and hospitality of the University of Salzburg and the Centre for Pure Mathematics in the School of Mathematics and Applied Statistics at the University of Wollongong.

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