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Distance Geometry in Quasihypermetric Spaces. III

Mathematische Nachrichten (Impact Factor: 0.68). 04/2011; 284(5-6). DOI: 10.1002/mana.200810216
Source: arXiv

ABSTRACT

Let (X, d) be a compact metric space and let denote the space of all finite signed Borel measures on X. Define by
and set where μ ranges over the collection of signed measures in 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 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. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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    • "It is straightforward to confirm that the linear span of these functions in C([a, b]) is exactly the subspace of piecewise linear continuous functions, which is dense in C([a, b]), and it follows that im T is dense in C([a, b]). Since M([a, b]) = (b − a)/2 < ∞ (see Lemma 3.5 of [5] or Corollary 3.2 of [27]), we have the following: "
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    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. The metric space $(X, d)$ is quasihypermetric if for all $n \in \N$, all $\alpha_1, ..., \alpha_n \in \R$ satisfying $\sum_{i=1}^n \alpha_i = 0$ and all $x_1, ..., x_n \in X$, one has $\sum_{i,j=1}^n \alpha_i \alpha_j d(x_i, x_j) \leq 0$. Without the quasihypermetric property $M(X)$ is infinite, while with the property a natural semi-inner product structure becomes available on $\mathcal{M}_0(X)$, the subspace of $\mathcal{M}(X)$ of all measures of total mass 0. This paper explores: operators and functionals which provide natural links between the metric structure of $(X, d)$, the semi-inner product space structure of $\mathcal{M}_0(X)$ and the Banach space $C(X)$ of continuous real-valued functions on $X$; conditions equivalent to the quasihypermetric property; the topological properties of $\mathcal{M}_0(X)$ with the topology induced by the semi-inner product, and especially the relation of this topology to the weak-$*$ topology and the measure-norm topology on $\mathcal{M}_0(X)$; and the functional-analytic properties of $\mathcal{M}_0(X)$ as a semi-inner product space, including the question of its completeness. A later paper [Peter Nickolas and Reinhard Wolf, Distance Geometry in Quasihypermetric Spaces. II] will apply the work of this paper to a detailed analysis of the constant $M(X)$.
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    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 measures in $\mathcal{M}(X)$ of total mass 1. The space $(X, d)$ is \emph{quasihypermetric} if $I(\mu) \leq 0$ for all measures $\mu$ in $\mathcal{M}(X)$ of total mass 0 and is \emph{strictly quasihypermetric} if in addition the equality $I(\mu) = 0$ holds amongst measures $\mu$ 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 [Peter Nickolas and Reinhard Wolf, \emph{Distance geometry in quasihypermetric spaces. I}, \emph{II} and \emph{III}].
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