Comparing the generalized roundness of metric spaces

TL;DR

This paper introduces a new method for comparing the generalized roundness of metric spaces, applying it to Banach spaces and metric trees, revealing new classes and properties of these spaces.

Contribution

It develops a novel technique based on finite representability for comparing generalized roundness, with applications to Banach spaces and metric trees, including new examples and embedding properties.

Findings

Ultrapowers and second duals share the same generalized roundness as the original Banach space.

No Banach space with positive generalized roundness is uniformly homeomorphic to c0 or ℓp for p>2.

Identified new classes of metric trees with generalized roundness one, including finite diameter trees.

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.