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Distance geometry in quasihypermetric spaces. I

Journal Article


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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

    $

    \mbar(X) = \sup \Imu,

    $

    where $\mu$ ranges over the collection of signed measures in $\MX$

    of total mass~$1$.

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

    for all $n \in \N$, all $\alpha_1, \ldots, \alpha_n \in \R$

    satisfying

    $\sum_{i=1}^n \alpha_i = 0$ and all $x_1, \ldots, x_n \in X$,

    one has

    $\sum_{i,j=1}^n \alpha_i \alpha_j d(x_i, x_j) \leq 0$.

    Without the quasihypermetric property $\mbar(X)$ is infinite,

    while with the property a natural semi-inner product structure

    becomes available on $\Mzero(X)$,

    the subspace of $\MX$ of all measures of total mass~$0$.

    This paper explores: operators and functionals

    which provide natural links between the metric structure

    of~$\Xd$, the semi-inner product space structure of $\Mzero(X)$

    and the Banach space~$C(X)$ of continuous real-valued functions

    on~$X$; conditions equivalent to the quasihypermetric property;

    the topological properties of $\Mzero(X)$ with the topology

    induced by the semi-inner product, and especially

    the relation of this topology to the \wstar{} topology

    and the measure-norm topology on $\Mzero(X)$;

    and the functional-analytic properties of $\Mzero(X)$

    as a semi-inner product space, including the question

    of its completeness.

    A~later paper [Peter Nickolas and Reinhard Wolf,

    \emph{Distance Geometry in Quasihypermetric Spaces.~II}]

    will apply the work of this paper to a detailed analysis

    of the constant $\mbar(X)$.

    \end{abstract}

Publication Date


  • 2009

Citation


  • Nickolas, P. & Nickolas, P. (2009). Distance geometry in quasihypermetric spaces. I. Bulletin of the Australian Mathematical Society, 80 (1), 1-25.

Scopus Eid


  • 2-s2.0-67149101171

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=10570&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/3234

Number Of Pages


  • 24

Start Page


  • 1

End Page


  • 25

Volume


  • 80

Issue


  • 1

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

    $

    \mbar(X) = \sup \Imu,

    $

    where $\mu$ ranges over the collection of signed measures in $\MX$

    of total mass~$1$.

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

    for all $n \in \N$, all $\alpha_1, \ldots, \alpha_n \in \R$

    satisfying

    $\sum_{i=1}^n \alpha_i = 0$ and all $x_1, \ldots, x_n \in X$,

    one has

    $\sum_{i,j=1}^n \alpha_i \alpha_j d(x_i, x_j) \leq 0$.

    Without the quasihypermetric property $\mbar(X)$ is infinite,

    while with the property a natural semi-inner product structure

    becomes available on $\Mzero(X)$,

    the subspace of $\MX$ of all measures of total mass~$0$.

    This paper explores: operators and functionals

    which provide natural links between the metric structure

    of~$\Xd$, the semi-inner product space structure of $\Mzero(X)$

    and the Banach space~$C(X)$ of continuous real-valued functions

    on~$X$; conditions equivalent to the quasihypermetric property;

    the topological properties of $\Mzero(X)$ with the topology

    induced by the semi-inner product, and especially

    the relation of this topology to the \wstar{} topology

    and the measure-norm topology on $\Mzero(X)$;

    and the functional-analytic properties of $\Mzero(X)$

    as a semi-inner product space, including the question

    of its completeness.

    A~later paper [Peter Nickolas and Reinhard Wolf,

    \emph{Distance Geometry in Quasihypermetric Spaces.~II}]

    will apply the work of this paper to a detailed analysis

    of the constant $\mbar(X)$.

    \end{abstract}

Publication Date


  • 2009

Citation


  • Nickolas, P. & Nickolas, P. (2009). Distance geometry in quasihypermetric spaces. I. Bulletin of the Australian Mathematical Society, 80 (1), 1-25.

Scopus Eid


  • 2-s2.0-67149101171

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=10570&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/3234

Number Of Pages


  • 24

Start Page


  • 1

End Page


  • 25

Volume


  • 80

Issue


  • 1