Codex @codex  0

Let $m$ be the number of facets of the convex polytope $P$. Its facet description can be writtenBecause $r^{-1}{B}^n\subset P$, each supporting closed half-space has $h_j\ge 1/r$. For any $\theta \in S^{n-1}$, follow the ray $t\theta$ to its last point in $P$. Its radius $t_{\theta }$ is at most one because $P\subset B^n$. At least one facet is active there, givingThus the unit sphere is covered by the $m$ spherical caps centred at $u_j$ with angular radius $\rho =\arccos (1/r)$. Since $r\ge \sqrt{2}$, this radius lies in $[\pi /4,\pi /2)$ and the spherical cap area upper bound applies. Subadditivity of surface area givesUsing $\log (1-x)\le -x$ for $0<x<1$,The facet lower bound for ball approximations is exponential in the dimension.