Approximation of convex bodies by polytopes with respect to minimal width and diameter
Volume 149 / 2017
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 what is the greatest number $\varLambda _n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ contains a polytope $P$ with at most $n$ vertices for which $\varLambda _n ({\mathcal K}^d) \leq {w(P)/w(C)}$. We give a lower estimate of $\varLambda _n ({\mathcal K}^d)$ for $n \geq 2d$ based on estimates of the smallest radius of $\lfloor {{n/2}} \rfloor $ antipodal pairs of spherical caps that cover the unit sphere of $E^d$. We show that $\varLambda _3 ({\mathcal K}^2) \geq {\frac 1 2}(3- \sqrt 3)$, and $\varLambda _n ({\mathcal K}^2) \geq \cos {\frac \pi {2 \lfloor {n/2} \rfloor }}$ for every $n \geq 4$. We also consider the dual question of estimating the smallest number $\Delta _n ({\mathcal K}^d)$ such that for every $C \in {\mathcal K}^d$ there exists a polytope $P\supset C$ with at most $n$ facets for which ${{\rm diam}(P)/{\rm diam}(C)} \leq \Delta _n ({\mathcal K}^d)$. 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 $n \geq 4$.
