  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 signed
measures in $\mathcal{M}(X)$ of total mass 1. This paper, with an earlier and a
subsequent paper [Peter Nickolas and Reinhard Wolf, Distance geometry in
quasihypermetric spaces. I and III], investigates the geometric constant $M(X)$
and its relationship to the metric properties of $X$ and the
functional-analytic properties of a certain subspace of $\mathcal{M}(X)$ when
equipped with a natural semi-inner product. Using the work of the earlier
paper, this paper explores measures which attain the supremum defining $M(X)$,
sequences of measures which approximate the supremum when the supremum is not
attained and conditions implying or equivalent to the finiteness of $M(X)$.
