Comparing the generalized 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 generalized roundness of metric spaces. We illustrate this technique in two different ways 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 generalized roundness as $X$, and (2) no Banach space of positive generalized roundness is uniformly homeomorphic to $c_{0}$ or $\ell_{p}$, $2 < p < \infty$. Our technique also leads to the identification of new classes of metric trees of generalized roundness one. In particular, we give the first examples of metric trees of generalized roundness one that have finite diameter. These results on metric trees provide a natural sequel to a paper of Caffarelli, Doust and Weston. In addition, we show that metric trees of generalized roundness one possess special Euclidean embedding properties that distinguish them from all other metric trees.