Finite Quasihypermetric Spaces

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 $I(mu) = int_X int_X d(x,y) dmu(x) dmu(y)$, 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}].

Distance Geometry in Quasihypermetric Spaces. III

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) dmu(x) dmu(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.

Distance Geometry in Quasihypermetric Spaces. II

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) dmu(x) dmu(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 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 $M(X)$.

Distance Geometry in Quasihypermetric Spaces. I

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) dmu(x) dmu(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 $M(X)$.