# Distance Geometry in Quasihypermetric Spaces. III - 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 two earlierpapers Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetricspaces. I and II, investigates the geometric constant $MX$ and itsrelationship to the metric properties of $X$ and the functional-analyticproperties of a certain subspace of $\mathcal{M}X$ when equipped with anatural semi-inner product. Specifically, this paper explores links between theproperties of $MX$ and metric embeddings of $X$, and the properties of $MX$when $X$ is a finite metric space.

Author: Peter Nickolas, Reinhard Wolf

Source: https://arxiv.org/