Distance geometry in quasihypermetric spaces. I

Authors

Peter Nickolas, University of WollongongFollow
Reinhard Wolf, Universitat Salzburg

RIS ID

29328

Publication Details

Nickolas, P. & Wolf, R. (2009). Distance geometry in quasihypermetric spaces. I. Bulletin of the Australian Mathematical Society, 80 (1), 1-25.

Abstract

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

and set M(X)=sup I(μ), where μ ranges over the collection of signed measures in (X) of total mass 1.

The metric space (X,d) is quasihypermetric if for all n, all α₁,…,α_n satisfying ∑ _i=1ⁿα_i=0 and all x₁,…,x_nX, the inequality ∑ _i,j=1ⁿα_iα_jd(x_i,x_j)≤0 holds. Without the quasihypermetric property M(X) is infinite, while with the property a natural semi-inner product structure becomes available on ₀(X), the subspace of (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 ₀(X) and the Banach space C(X) of continuous real-valued functions on X; conditions equivalent to the quasihypermetric property; the topological properties of ₀(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 ₀(X); and the functional-analytic properties of ₀(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).