On the Many Faces of Easily Covered Polytopes

Assume that $r{B}_2^n\subset P$ for some polytope $P\subset {\mathbb{R}}^n$, where $r\in (\frac{1}{2},1]$. Denote by $\mathcal{F}$ the set of facets of $P$, and by $N=N(P,B_2^n)$ the covering number of $P$ by the Euclidean unit ball $B_2^n$. We prove that if $\log N\le \frac{n}{8}$, then $$\mathrm{\mid }\mathcal{F}\mathrm{\mid }\ge {\left(\frac{1}{2(1-r\sqrt{1-\frac{4\log N}{n}})}\right)}^{\frac{n-1}{2}}\mathrm{.}$$