# Distance Geometry in Quasihypermetric Spaces. II - Mathematics > Metric Geometry

Abstract: Let $X, d$ be a compact metric space and let $\mathcal{M}X$ denote thespace of all finite signed Borel measures on $X$. Define $I \colon\mathcal{M}X \to \R$ by \ I\mu = \int X \int X dx,y d\mux d\muy, \and set $MX = \sup I\mu$, where $\mu$ ranges over the collection of signedmeasures in $\mathcal{M}X$ of total mass 1. This paper, with an earlier and asubsequent paper Peter Nickolas and Reinhard Wolf, Distance geometry inquasihypermetric spaces. I and III, investigates the geometric constant $MX$and its relationship to the metric properties of $X$ and thefunctional-analytic properties of a certain subspace of $\mathcal{M}X$ whenequipped with a natural semi-inner product. Using the work of the earlierpaper, this paper explores measures which attain the supremum defining $MX$,sequences of measures which approximate the supremum when the supremum is notattained and conditions implying or equivalent to the finiteness of $MX$.

Author: Peter Nickolas, Reinhard Wolf

Source: https://arxiv.org/