DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. I

Abstract

AbstractLet (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).