Cardinality of $\Lambda$ Determines the Geometry of $\mathsf{B}_{\ell_\infty(\Lambda)}$ and $\mathsf{B}_{\ell_\infty(\Lambda)^*}$

We study the geometry of the unit ball of $\ell_\infty(\Lambda)$ and of the dual space, proving, among other things, that $\Lambda$ is countable if and only if $1$ is an exposed point of $\mathsf{B}_{\ell_\infty(\Lambda)}$. On the other hand, we prove that $\Lambda$ is finite if and only if the $\delta_\lambda$ are the only functionals taking the value $1$ at a canonical element and vanishing at all other canonical elements. We also show that the restrictions of evaluation functionals to a $2$-dimensional subspace are not necessarily extreme points of the dual of that subspace. Finally, we prove that if $\Lambda$ is uncountable, then the face of $\mathsf{B}_{\ell_\infty(\Lambda)^*}$ consisting of norm $1$ functionals attaining their norm at the constant function $1$ has empty interior relative to $\mathsf{S}_{\ell_\infty(\Lambda)^*}$.