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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}].

Publication Date


  • 2009

Citation


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

Scopus Eid


  • 2-s2.0-70350031272

Ro Metadata Url


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

Number Of Pages


  • 19

Start Page


  • 243

End Page


  • 262

Volume


  • 124

Issue


  • 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}].

Publication Date


  • 2009

Citation


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

Scopus Eid


  • 2-s2.0-70350031272

Ro Metadata Url


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

Number Of Pages


  • 19

Start Page


  • 243

End Page


  • 262

Volume


  • 124

Issue


  • 3