Distance Geometry in Quasihypermetric Spaces. I
| dc.creator | Nickolas, Peter | |
| dc.creator | Wolf, Reinhard | |
| dc.date | 2008-09-04 | |
| dc.date.accessioned | 2026-07-07T10:00:38Z | |
| dc.date.available | 2026-07-07T10:00:38Z | |
| dc.description | 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. The metric space $(X, d)$ is quasihypermetric if for all $n \in \N$, all $α_1, ..., α_n \in \R$ satisfying $\sum_{i=1}^n α_i = 0$ and all $x_1, ..., x_n \in X$, one has $\sum_{i,j=1}^n α_i α_j d(x_i, x_j) \leq 0$. 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 $M(X)$. | |
| dc.description | To appear in the Bulletin of the Australian Mathematical Society; 25 pages. References [27] and [28] are arXiv:0809.0744v1 [math.MG] and arXiv:0809.0746v1 [math.MG] | |
| dc.identifier | https://arxiv.org/abs/0809.0740 | |
| dc.identifier | http://arxiv.org/abs/0809.0740 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/168372 | |
| dc.subject | Metric Geometry | |
| dc.subject | 51K05 (Primary), 54E45, 31C45 (Secondary) | |
| dc.title | Distance Geometry in Quasihypermetric Spaces. I | |
| dc.type | text |