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](https://math.stackexchange.com/questions/1273842/regular-n-gon-between-two-concentric-circles) 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?

**Edit:** Poking around at tangentially-related questions (e.g. [1](https://mathoverflow.net/questions/429633/bounding-the-number-of-facets-of-a-polytope-to-approximate-a-given-convex-shape), [2](https://mathoverflow.net/questions/429332/d-ball-approximation-for-d-gg-1-with-a-convex-hull-of-random-points-on-its-b?rq=1), [3](https://mathoverflow.net/questions/433967/approximating-any-convex-shape-in-mathbbrd-with-a-polytope-having-mathrm?noredirect=1&lq=1)), maybe this is a *lot* harder of a question than I think.

**Edit 2:** I will add that I am also interested in known results for low-dimensions (particularly $\mathbb R^3$), and also if there are any existing methods for constructing such polytopes *ad hoc* in $\mathbb R^d$ for a particular $d$.