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# Distance Geometry in Quasihypermetric Spaces. III

10/2008;
Source: arXiv

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 $I \colon \mathcal{M}(X) \to \R$ by $I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y),$ and set $M(X) = \sup I(\mu)$, where $\mu$ ranges over the collection of signed measures in $\mathcal{M}(X)$ 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 $\mathcal{M}(X)$ 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.

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##### Article:Distance geometry in quasihypermetric spaces. II
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ABSTRACT: Let (X, d) be a compact metric space and let denote the space of all finite signed Borel measures on X. Define by I(μ) = ∫X∫Xd(x, y) dμ(x)dμ(y), and set , where μ ranges over the collection of signed measures in of total mass 1. This paper, with an earlier and a subsequent paper [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and III], 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 when equipped with a natural semi-inner product. Using the work of the earlier paper, this paper explores measures which attain the supremum defining M(X), sequences of measures which approximate the supremum when the supremum is not attained and conditions implying or equivalent to the finiteness of M(X). © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Mathematische Nachrichten 01/2011; 284(2‐3):332 - 341. · 0.68 Impact Factor
• Extremal problems of distance geometry related to energy integrals. Ralph Alexander, Kenneth B Stolarsky . 1974. Trans. Amer. Math. Soc 193 1-31.
• Sur les inégalités valides dans L 1. Patrice Assouad, Un, Acad, Sci . 1977. Paris Sér. A-B European J. Combin 2853 361-363.

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### Keywords

certain subspace

compact metric space

Define $I \colon \mathcal{M}(X) finite finite metric space geometric constant$M(X)\$

metric embeddings

metric properties

natural semi-inner product

quasihypermetric spaces

Reinhard Wolf

total mass 1