Lemma: Polytope approximation

Proof. The upper inclusion follows from the length bound. For unit $u$, choose a long generator of direction within ${\delta }_n$ of $u$. Then $\Vert s/L_n-u{\Vert }_2\le 2{\delta }_n$, so the support function of $P_n$ in direction $u$ is at least $1-2{\delta }_n$. The supporting-half-space description of a convex body gives the lower inclusion. Radial projection gives the Hausdorff bound.