# Finite quasihypermetric spaces

Journal Article

### Abstract

• Let $\Xd$ be a compact metric space and let

$\MX$ denote the space of all finite signed Borel measures on~$X$.

Define $I \colon \MX \to \R$ by

$\Imu = \int_X \! \int_X d(x,y) \, d\mu(x) d\mu(y)$,

and set $M(X) = \sup \Imu$, where $\mu$ ranges over

the collection of measures in~$\MX$ of total mass~$1$.

The space $\Xd$ is \textit{quasihypermetric} if

$I(\mu) \leq 0$ for all measures~$\mu$ in $\M(X)$

of total mass~$0$ and is \textit{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}].

• 2009

### Citation

• Nickolas, P. & Wolf, R. (2009). Finite quasihypermetric spaces. Acta Mathematica Hungarica, 124 (3), 243-262.

### Scopus Eid

• 2-s2.0-70350031272

• http://ro.uow.edu.au/infopapers/3235

• 19

• 243

• 262

• 124

• 3

### Abstract

• Let $\Xd$ be a compact metric space and let

$\MX$ denote the space of all finite signed Borel measures on~$X$.

Define $I \colon \MX \to \R$ by

$\Imu = \int_X \! \int_X d(x,y) \, d\mu(x) d\mu(y)$,

and set $M(X) = \sup \Imu$, where $\mu$ ranges over

the collection of measures in~$\MX$ of total mass~$1$.

The space $\Xd$ is \textit{quasihypermetric} if

$I(\mu) \leq 0$ for all measures~$\mu$ in $\M(X)$

of total mass~$0$ and is \textit{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}].

• 2009

### Citation

• Nickolas, P. & Wolf, R. (2009). Finite quasihypermetric spaces. Acta Mathematica Hungarica, 124 (3), 243-262.

### Scopus Eid

• 2-s2.0-70350031272

• http://ro.uow.edu.au/infopapers/3235

• 19

• 243

• 262

• 124

• 3