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

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 $\Lambda_n ({\mathcal K}^d) \leq \frac{w(P)}{w(C)}$. We give a lower estimate of $\Lambda_n ({\mathcal K}^d)$ for $n \geq 2d$ based on estimates of the smallest radius of $\big\lfloor {\frac{n}{2}} \big\rfloor$ antipodal pairs of spherical caps that cover the unit sphere of $E^d$. We show that $\Lambda_3 ({\mathcal K}^2) \geq {\frac 1 2}(3- \sqrt 3)$, and $\Lambda_n ({\mathcal K}^2) \geq \cos {\frac \pi {2 \lfloor {n/2} \rfloor}}$ for every $n \geq 4$. 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 $\frac{{\rm diam}(P)}{{\rm diam}(C)} \leq \Delta_n ({\mathcal K}^d)$. We give an upper bound of $\Delta_n ({\mathcal K}^d)$ for $n \geq 2d$. In particular, $\Delta_n ({\mathcal K}^2) \leq 1/ \cos {\frac \pi {2 \lfloor {n/2} \rfloor}}$ for $n \geq 4$.