Distance geometry in quasihypermetric spaces. I

Let $\Xd$ be a compact metric space and let

$\MX$ denote the space of all finite signed Borel measures on~$X$.

Define $I \colon \MX \to \R$ by

\[

\Imu = \int_X \! \int_X d(x,y) \, d\mu(x) d\mu(y),

\]

and set

$

\mbar(X) = \sup \Imu,

$

where $\mu$ ranges over the collection of signed measures in $\MX$

of total mass~$1$.

The metric space $\Xd$ is \textit{quasihypermetric} if

for all $n \in \N$, all $\alpha_1, \ldots, \alpha_n \in \R$

satisfying

$\sum_{i=1}^n \alpha_i = 0$ and all $x_1, \ldots, x_n \in X$,

one has

$\sum_{i,j=1}^n \alpha_i \alpha_j d(x_i, x_j) \leq 0$.

Without the quasihypermetric property $\mbar(X)$ is infinite,

while with the property a natural semi-inner product structure

becomes available on $\Mzero(X)$,

the subspace of $\MX$ of all measures of total mass~$0$.

This paper explores: operators and functionals

which provide natural links between the metric structure

of~$\Xd$, the semi-inner product space structure of $\Mzero(X)$

and the Banach space~$C(X)$ of continuous real-valued functions

on~$X$; conditions equivalent to the quasihypermetric property;

the topological properties of $\Mzero(X)$ with the topology

induced by the semi-inner product, and especially

the relation of this topology to the \wstar{} topology

and the measure-norm topology on $\Mzero(X)$;

and the functional-analytic properties of $\Mzero(X)$

as a semi-inner product space, including the question

of its completeness.

A~later paper [Peter Nickolas and Reinhard Wolf,

\emph{Distance Geometry in Quasihypermetric Spaces.~II}]

will apply the work of this paper to a detailed analysis

of the constant $\mbar(X)$.

\end{abstract}