Distance geometry in quasihypermetric spaces. II

Abstract

Abstract. 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 byand set M (X) = sup I(µ), where µ ranges over the collection of signed measures in M(X) of total mass 1. This paper, with an earlier and a subsequent paper [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and III ], investigates the geometric constant M (X) and its relationship to the metric properties of X and the functional-analytic properties of a certain subspace of M(X) when equipped with a natural semi-inner product. Using the work of the earlier paper, this paper explores measures which attain the supremum defining M (X), sequences of measures which approximate the supremum when the supremum is not attained and conditions implying or equivalent to the finiteness of M (X).

Estimating the gap of finite metric spaces of strict p-negative type

“…The following result was shown in [13] (Theorem 3.5) for general compact metric spaces of 1-negative type. For completeness we add the proof now given in terms of matrices.…”

“…Definition 6.1. (see Theorem 3.5 of [13])Let (X 1 , d 1 ) and (X 2 , d 2 ) be two finite metric spaces such that X 1 ∩ X 2 = ∅. Further let c > 0 with 2c ≥ max(∆(X 1 ), ∆(X 2 )).…”

“…The above defined p-negative type gap Γ(X, p) can be used to enlarge the p-parameter, for wich a given finite metric space is of p-negative type: It is shown in [9] (Theorem 3.3) that a finite metric space (X, d) with cardinality n = |X| ≥ 3 of strict p-negative type is of strict q-negative type for all q ∈ For basic information on p-negative type spaces (1-negative type spaces are also known as quasihypermetric spaces) see for example [2,8,9,12,13,14,15,16,18].…”

“…The constant M p was studied in [12,13,14,15] for general compact metric spaces of 1negative type, it also appears in the study of Euclidean distance matrices (see Corollary 3.2 in [7]). Following the ideas of these papers we obtain Theorem 3.7.…”

“…Following the ideas of these papers we obtain Theorem 3.7. (Compare to Theorem 3.1 in [12]) Let (X, d) be a finite metric space of at least two points. If M p < ∞ then (X, d) is of p-negative type.…”