Abstract

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 semiinner 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 semiinner product space structure of $\Mzero(X)$
and the Banach space~$C(X)$ of continuous realvalued functions
on~$X$; conditions equivalent to the quasihypermetric property;
the topological properties of $\Mzero(X)$ with the topology
induced by the semiinner product, and especially
the relation of this topology to the \wstar{} topology
and the measurenorm topology on $\Mzero(X)$;
and the functionalanalytic properties of $\Mzero(X)$
as a semiinner 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}