# Distance Geometry in Quasihypermetric Spaces. I - 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.The metric space $X, d$ is quasihypermetric if for all $n \in \N$, all$\alpha 1, ., \alpha n \in \R$ satisfying $\sum {i=1}^n \alpha i = 0$ and all$x 1, ., x n \in X$, one has $\sum {i,j=1}^n \alpha i \alpha j dx i, x j\leq 0$. Without the quasihypermetric property $MX$ is infinite, while withthe property a natural semi-inner product structure becomes available on$\mathcal{M} 0X$, the subspace of $\mathcal{M}X$ of all measures of totalmass 0. This paper explores: operators and functionals which provide naturallinks between the metric structure of $X, d$, the semi-inner product spacestructure of $\mathcal{M} 0X$ and the Banach space $CX$ of continuousreal-valued functions on $X$; conditions equivalent to the quasihypermetricproperty; the topological properties of $\mathcal{M} 0X$ with the topologyinduced by the semi-inner product, and especially the relation of this topologyto the weak-$*$ topology and the measure-norm topology on $\mathcal{M} 0X$;and the functional-analytic properties of $\mathcal{M} 0X$ as a semi-innerproduct space, including the question of its completeness. A later paper PeterNickolas and Reinhard Wolf, Distance Geometry in Quasihypermetric Spaces. IIwill apply the work of this paper to a detailed analysis of the constant$MX$.

Author: Peter Nickolas, Reinhard Wolf

Source: https://arxiv.org/