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COMPARING THE GENERALISED ROUNDNESS OF METRIC SPACES
Published online by Cambridge University Press: 02 November 2016
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.
MSC classification
Primary:
46B07: Local theory of Banach spaces
- Type
- Research Article
- Information
- 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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