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COMPARING THE GENERALISED ROUNDNESS OF METRIC SPACES

Published online by Cambridge University Press:  02 November 2016

LUKIEL LEVY-MOORE
Affiliation:
Department of Economics, New York University, New York, NY 10012, USA email LukielLevyMoore@nyu.edu
MARGARET NICHOLS
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA email mnichols@math.uchicago.edu
ANTHONY WESTON*
Affiliation:
Department of Mathematics and Statistics, Canisius College, Buffalo, NY 14208, USA Department of Decision Sciences, University of South Africa, Unisa 0003, South Africa email westona@canisius.edu, westoar@unisa.ac.za
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Abstract

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Motivated by the local theory of Banach spaces, we introduce a notion of finite representability for metric spaces. This allows us to develop a new technique for comparing the generalised roundness of metric spaces. We illustrate this technique by applying it to Banach spaces and metric trees. In the realm of Banach spaces we obtain results such as the following: (1) if ${\mathcal{U}}$ is any ultrafilter and $X$ is any Banach space, then the second dual $X^{\ast \ast }$ and the ultrapower $(X)_{{\mathcal{U}}}$ have the same generalised roundness as $X$, and (2) no Banach space of positive generalised roundness is uniformly homeomorphic to $c_{0}$ or $\ell _{p}$, $2<p<\infty$. For metric trees, we give the first examples of metric trees of generalised roundness one that have finite diameter. In addition, we show that metric trees of generalised roundness one possess special Euclidean embedding properties that distinguish them from all other metric trees.

Type
Research Article
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

The research in this paper was initiated at the 2011 Cornell University Summer Mathematics Institute (NSF grant DMS-0739338) and completed at the University of South Africa (Unisa). The second named author was partially supported by the National Science Foundation Graduate Research Fellowship Program (NSF grant DGE-1144082).

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