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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 at most $n$ 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?

pyridoxal_trigeminus
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    To minimize $n$, you need to have facets whose vertices are all on the outer sphere, and which are tangent to the inner sphere. If you assume the facets are regular (the main polytope may not be regular, but I think it very likely that the facets will be), this should be enough to calculate the maximum size of the facet, and therefore the hyper-solid-angle that it subtends. Dividing the full $d$-solid-angle of the $d$-sphere by this $d$-solid-angle of the facet gives you a lower bound on $n$. – Paul Sinclair Mar 23 '23 at 23:53
  • Thanks @PaulSinclair - I'm offering a bounty for a more complete answer. – pyridoxal_trigeminus Mar 25 '23 at 06:33

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