  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.
