Let (X, d) be a compact metric space and let M(X) denote the space of all ﬁnite signed Borel measures on X. Define I : M(X) → R by I(µ) = \int_X \int_X d(x, y) dµ(x) dµ(y), and set M(X) = sup I(µ), where µ ranges over the collection of signed measures in M(X) of total mass 1. This paper, with two earlier papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and II ], 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 M(X) when equipped with a natural semi-inner product. Speciﬁcally, this paper explores links between the properties of M(X ) and metric embeddings of X, and the properties of M(X) when X is a ﬁnite metric space.