  The maximal roundness of a metric space is a quantity that arose in the study
of embeddings and renormings. In the setting of Banach spaces, it was shown by
Enflo that roundness takes on a much simpler form. In this paper we provide
simple computations of the roundness of many standard Banach spaces, such as
$\ell^{p}$, the Lebesgue-Bochner spaces $\ell^{p}(\ell^{q})$ and the Schatten
ideals $S_{p}$. We also introduce a property that is dual to that of roundness,
which we call coroundness, and make explicit the relation of these properties
to the geometric concepts of smoothness and convexity of Banach spaces.
Building off the work of Enflo, we are then able to provide multiple
non-trivial equivalent conditions for a Banach space to possess maximal
roundness greater than $1$. Using these conditions, we are able to conclude
that certain Orlicz spaces possess non-trivial values of roundness and
coroundness. Finally, we also use these conditions to provide an explicit
example of a $2$-dimensional Banach space whose maximal roundness is not equal
to that of its dual.
