Bulletin of the Australian Mathematical Society

Table of Contents - August 2009 - Volume 80, Issue 01  

Research Article

DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. I

PETER NICKOLASa1 c1 and REINHARD WOLFa2

a1 School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia (email: peter_nickolas@uow.edu.au)

a2 Institut für Mathematik, Universität Salzburg, Hellbrunnerstrasse 34, A-5020 Salzburg, Austria (email: Reinhard.Wolf@sbg.ac.at)

Abstract

Let (X,d) be a compact metric space and let xs2133(X) denote the space of all finite signed Borel measures on X. Define I:xs2133(X)→xs211D by

\[ I(\mu ) = \int _X \! \int _X d(x,y) \,d\mu (x) \,d\mu (y), \]

and set M(X)=sup I(μ), where μ ranges over the collection of signed measures in xs2133(X) of total mass 1. The metric space (X,d) is quasihypermetric if for all nxs2208xs2115, all α1,…,αnxs2208xs211D satisfying ∑ i=1nαi=0 and all x1,…,xnxs2208X, 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 xs21330(X), the subspace of xs2133(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 xs21330(X) and the Banach space C(X) of continuous real-valued functions on X; conditions equivalent to the quasihypermetric property; the topological properties of xs21330(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 xs21330(X); and the functional-analytic properties of xs21330(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).

(Received March 10 2008)

2000 Mathematics subject classification

  • primary 51K05; secondary 54E45;
  • 31C45

Keywords and phrases

  • compact metric space;
  • finite metric space;
  • quasihypermetric space;
  • metric embedding;
  • signed measure;
  • signed measure of mass zero;
  • spaces of measures;
  • distance geometry;
  • geometric constant

Correspondence:

c1 For correspondence; e-mail: peter˙nickolas@uow.edu.au

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.