Distance Geometry in Quasihypermetric Spaces. II

TL;DR

This paper investigates the geometric constant M(X) in compact metric spaces, analyzing measures that attain or approximate its supremum and conditions for its finiteness, linking metric and functional-analytic properties.

Contribution

It extends previous work by characterizing measures that attain or approximate M(X) and establishing conditions for its finiteness in quasihypermetric spaces.

Findings

Identifies measures that attain the supremum M(X).

Describes sequences of measures approximating M(X).

Provides conditions equivalent to the finiteness of M(X).

Abstract

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[ I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y), \] and set $M(X) = \sup I(\mu)$, where $\mu$ ranges over the collection of signed measures in $\mathcal{M}(X)$ 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 $\mathcal{M}(X)$ 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 $M(X)$.