COMPARING THE GENERALISED ROUNDNESS OF METRIC SPACES

Abstract

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.