Abstract
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 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 [11] [13].
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Nickolas, P., Wolf, R. Finite quasihypermetric spaces. Acta Math Hung 124, 243–262 (2009). https://doi.org/10.1007/s10474-009-8182-2
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DOI: https://doi.org/10.1007/s10474-009-8182-2
Key words and phrases
- compact metric space
- finite metric space
- quasihypermetric space
- metric embedding
- signed measure
- signed measure of mass zero
- spaces of measures
- distance geometry
- geometric constant