Roundness properties of Banach spaces

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