 Facet lower bound for ball approximations

If $r\ge \sqrt{2}$ and a convex polytope satisfies $r^{-1}{B}^n\subseteq P\subseteq B^n$, then it has at least $(1-r^{-2}{)}^{-n/2}\ge e^{n/(2r^2)}$ facets. Its outward facet normals define spherical caps covering the unit sphere; the spherical cap area upper bound supplies the estimate.