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Acta Mathematica Hungarica

, Volume 124, Issue 3, pp 243–262 | Cite as

Finite quasihypermetric spaces

  • P. Nickolas
  • R. Wolf
Article

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].

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 

2000 Mathematics Subject Classification

primary 51K05 secondary 54E45, 31C45 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2009

Authors and Affiliations

  1. 1.School of Mathematics and Applied StatisticsUniversity of WollongongWollongongAustralia
  2. 2.Institut für MathematikUniversität SalzburgSalzburgAustria

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