Distance geometry in quasihypermetric spaces. III

Abstract

Key words Compact metric space, finite metric space, quasihypermetric space, metric embedding, signed measure, signed measure of mass zero, spaces of measures, distance geometry, geometric constant MSC (2010) Primary: 51K05; Secondary: 54E45, 31C45Let (X, d) be a compact metric space and let M(X) denote the space of all finite signed Borel measures on X.and 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 two earlier papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and II], 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. Specifically, this paper explores links between the properties of M (X) and metric embeddings of X, and the properties of M (X) when X is a finite metric space.

A note on the metric geometry of the unit ball

“…and D t is isometric with a line segment of length 2, so by Corollary 3.2 of [10] we have I (μπ −1 t ) ≤ 1, and hence…”

“…Since the functional I is weak- * continuous on M + 1 ([−1, 1]) (see, for example, Corollary 2.8 of [9]), we finally have…”

“…As a functional on subsets of Euclidean spaces, M + was studied in detail by Björck [3]; see also [11].…”

“…The study of the geometric constant M(X ) for general X presents many interesting features and subtleties; a simple illustration of this is that M(X ) may be infinite even for finite X . In [9][10][11] and [12], the second and third authors have made a detailed study of certain aspects of the geometry of compact metric spaces X , with emphasis on the properties of M(X ). A few results from these papers will be referred to below.…”

Finite quasihypermetric spaces

“…In [11] and [12], we developed a general framework within which to discuss aspects of the geometry of a compact quasihypermetric or strictly quasihypermetric space X, with special emphasis on the behaviour of the geometric constant M (X). In [13], we investigated within this framework the role of metric embeddings in the theory and some of the properties of nite metric spaces.…”