This is related to a research problem that is interested in approximation of spheres by convex polytopes.

Let $C_r$ and $C_R$ be two spheres in $\mathbb R^d$ of radius $r$ and $R$, respectively, where $R>r$. WLOG assume they are centered at the origin.

I would like to know the smallest $n$ (or a lower bound on $n$) such that a convex polytope with $n$ facets will fit between $C_r$ and $C_R$, i.e., a polytope $P_n$ such that $C_r \subset P_n \subset C_R$.

This is a generalization of this question from circles in the plane to spheres of arbitrary dimension.

Is there a straightforward way to extend the result in $\mathbb R^2$ to $\mathbb R^d$, or do similar results exist already?