Finite Quasihypermetric Spaces
| dc.creator | Nickolas, Peter | |
| dc.creator | Wolf, Reinhard | |
| dc.date | 2009-02-25 | |
| dc.date.accessioned | 2026-07-07T12:47:04Z | |
| dc.date.available | 2026-07-07T12:47:04Z | |
| 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 $I(mu) = \int_X \int_X d(x,y) dμ(x) dμ(y)$, and set $M(X) = \sup I(mu)$, where $μ$ ranges over the collection of measures in $\mathcal{M}(X)$ of total mass 1. The space $(X, d)$ is \emph{quasihypermetric} if $I(μ) \leq 0$ for all measures $μ$ in $\mathcal{M}(X)$ of total mass 0 and is \emph{strictly quasihypermetric} if in addition the equality $I(μ) = 0$ holds amongst measures $μ$ 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}]. | |
| dc.description | 21 pages. References [11], [12] and [13] are arXiv:0809.0740v1 [math.MG], arXiv:0809.0744v1 [math.MG] and arXiv:0809.0746v1 [math.MG], resp | |
| dc.identifier | https://arxiv.org/abs/0902.4483 | |
| dc.identifier | http://arxiv.org/abs/0902.4483 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/221592 | |
| dc.subject | Metric Geometry | |
| dc.subject | 51K05; 54E45, 31C45 | |
| dc.title | Finite Quasihypermetric Spaces | |
| dc.type | text |