DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. I.

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 I (μ) = ∫X∫X d(x, y) dμ(x) dμ(y), 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 ϵ ℕ, all α1,…, …n ϵ ℝ satisfying ∑i=1n αi and all x1,…,xn ϵ X, the inequality ∑i,j=1n αiαjd(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 semiinner 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). [ABSTRACT FROM AUTHOR]