Approximation of convex bodies by polytopes with respect to minimal width and diameter

Abstract

Denote by ${\mathcal{K}}^d$ the family of convex bodies in $E^d$ and by $w(C)$ the minimal width of $C\in {\mathcal{K}}^d$. We ask for the greatest number ${\Lambda }_n({\mathcal{K}}^d)$ such that every $C\in {\mathcal{K}}^d$ contains a polytope $P$ with at most $n$ vertices for which ${\Lambda }_n({\mathcal{K}}^d)\le \frac{w(P)}{w(C)}$. We give a lower estimate of ${\Lambda }_n({\mathcal{K}}^d)$ for $n\ge 2d$ based on estimates of the smallest radius of $\lfloor \frac{n}{2}\rfloor$ antipodal pairs of spherical caps that cover the unit sphere of $E^d$. We show that ${\Lambda }_3({\mathcal{K}}^2)\ge \frac{1}{2}(3-\sqrt{3})$, and ${\Lambda }_n({\mathcal{K}}^2)\ge \cos \frac{\pi }{2\lfloor n/2\rfloor }$ for every $n\ge 4$. We also consider the dual question of estimating the smallest number ${\Delta }_n({\mathcal{K}}^d)$ such that every $C\in {\mathcal{K}}^d$ there exists a polytope $P\supset C$ with at most $n$ facets for which $\frac{\mathrm{diam}(P)}{\mathrm{diam}(C)}\le {\Delta }_n({\mathcal{K}}^d)$. We give an upper bound of ${\Delta }_n({\mathcal{K}}^d)$ for $n\ge 2d$. In particular, ${\Delta }_n({\mathcal{K}}^2)\le 1/\cos \frac{\pi }{2\lfloor n/2\rfloor }$ for $n\ge 4$.