  Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the
space of all finite signed Borel measures on $X$. Define $I \colon
\mathcal{M}(X) \to \R$ by $I(mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y)$, and
set $M(X) = \sup I(mu)$, where $\mu$ ranges over the collection of measures in
$\mathcal{M}(X)$ of total mass 1. The space $(X, d)$ is \emph{quasihypermetric}
if $I(\mu) \leq 0$ for all measures $\mu$ in $\mathcal{M}(X)$ of total mass 0
and is \emph{strictly quasihypermetric} if in addition the equality $I(\mu) =
0$ holds amongst measures $\mu$ of mass 0 only for the zero measure.
  This paper explores the constant $M(X)$ and other geometric aspects of $X$ in
the case when the space $X$ is finite, focusing first on the significance of
the maximal strictly quasihypermetric subspaces of a given finite
quasihypermetric space and second on the class of finite metric spaces which
are $L^1$-embeddable. While most of the results are for finite spaces, several
apply also in the general compact case. The analysis builds upon earlier more
general work of the authors [Peter Nickolas and Reinhard Wolf, \emph{Distance
geometry in quasihypermetric spaces. I}, \emph{II} and \emph{III}].
