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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?

Edit: Poking around at tangentially-related questions (e.g. 1, 2, 3), 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 $d$.

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    $\begingroup$ It is equivalent to asking how many spherical caps on the outer sphere whose cutting planes touch the inner one (so they are all of one fixed size if the spheres are concentric) are needed to cover the outer sphere completely. The trivial lower bound is the ratio of areas and it is correct up to a factor like $\text{const }d\log d$ with the idea to take a random collection of slightly shrunk caps, computing the area of the remaining portion, observing that it is so small that no noticeable size ball can fit into it, and expanding caps back (Rogers covering theorem). $\endgroup$
    – fedja
    Commented Mar 23, 2023 at 0:42
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    $\begingroup$ Nothing better is known in general as far as I know though there are some tighter bounds in low dimensions and computer algorithms for finding a nearly optimal cover when the area ratio is not too large. $\endgroup$
    – fedja
    Commented Mar 23, 2023 at 0:43
  • $\begingroup$ @fedja Do you have any references for the low-dimensional results you speak of? $\endgroup$
    – pyridoxal_trigeminus
    Commented Mar 23, 2023 at 1:28

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