Finite quasihypermetric spaces.

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: $$ \mathcal{M} $$( X) → ℝ by I(μ) = ∫ X∫ X d( x, y) dμ(x)dμ( y), and set M( X) = sup I(μ), where μ ranges over the collection of measures in $$ \mathcal{M} $$( X) of total mass 1. The space ( X, d) is quasihypermetric if I(μ) ≦ 0 for all measures μ in $$ \mathcal{M} $$( X) of total mass 0 and is strictly quasihypermetric if in addition the equality I(μ) = 0 holds amongst measures μ 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 L1-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 [11] [13]. [ABSTRACT FROM AUTHOR]