Finite quasihypermetric spaces |
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Authors: | P Nickolas R Wolf |
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Institution: | 1.School of Mathematics and Applied Statistics,University of Wollongong,Wollongong,Australia;2.Institut für Mathematik,Universit?t Salzburg,Salzburg,Austria |
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Abstract: | Let (X, d) be a compact metric space and let (X) denote the space of all finite signed Borel measures on X. Define I: (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 (X) of total mass 1. The space (X, d) is quasihypermetric if I(μ) ≦ 0 for all measures μ in (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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Keywords: | and phrases" target="_blank"> 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 |
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