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arXiv · 0809.0744

Distance Geometry in Quasihypermetric Spaces. II

Peter Nickolas·Reinhard Wolf
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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 \[ I(μ) = \int_X \int_X d(x,y) dμ(x) dμ(y), \] and set $M(X) = \sup I(μ)$, where $μ$ ranges over the collection of signed measures in $\mathcal{M}(X)$ 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 $\mathcal{M}(X)$ 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 $M(X)$.

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Peter Nickolas, Reinhard Wolf. 2008-09-04. Distance Geometry in Quasihypermetric Spaces. II. https://arxiv.org/abs/0809.0744

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