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.

度量空间的最大圆度是在嵌入和重新归一研究中出现的一个量。在 Banach 空间的设置中，Enflo 表明圆度采用了更简单的形式。在本文中，我们提供了许多标准 Banach 空间的圆度的简单计算，例如${\ell }^{磷}$, Lebesgue-Bochner 空间 ${\ell }^{磷}({\ell }^q)$ 和 Schatten 理想 ${秒}_{磷}$. 我们还引入了一个与圆度对偶的属性，我们称之为共圆度，并明确了这些属性与 Banach 空间的光滑度和凸度的几何概念的关系。基于 Enflo 的工作，我们可以为 Banach 空间提供多个非平凡等价条件，使其拥有大于 1 的最大圆度。使用这些条件，我们可以得出结论，某些 Orlicz 空间具有圆度和圆度。最后，我们还使用这些条件提供了一个二维 Banach 空间的显式示例，其最大圆度不等于其对偶的圆度。