On the Distance of Polytopes with Few Vertices to the Euclidean Ball

Abstract

Let $$n,N$$n,N be natural numbers satisfying $$n+1\le N\le 2n, B_2^n$$n+1≤N≤2n,B2n be the unit Euclidean ball in $${\mathbb R}^n$$Rn, and let $$P\subset B_2^n$$P?B2n be a convex $$n$$n-dimensional polytope with $$N$$N vertices and the origin in its interior. We prove that $$\begin{aligned} \inf \{\lambda \ge 1:\,B_2^n\subset \lambda P\}\ge cn/\sqrt{N-n}, \end{aligned}$$inf{??1:B2n??P}?cn/N-n,where $$c>0$$c>0 is a universal constant. As an immediate corollary, for any covering of $$S^{n-1}$$Sn-1 by $$N$$N spherical caps of geodesic radius $$\phi $$?, we get that $$\cos \phi \le C\sqrt{N-n}/n$$cos?≤CN-n/n for an absolute constant $$C>0$$C>0. Both estimates are optimal up to the constant multiples $$c, C$$c,C.