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.

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 $d$.

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?

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.