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

Marek Lassak¹

¹ Institute of Mathematics and Physics University of Technology and Life Sciences 85-789 Bydgoszcz, Poland

Colloquium Mathematicum, Tome 149 (2017) no. 1, pp. 21-32.

Résumé

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 $n \geq 4$.

Détail

DOI : 10.4064/cm6856-7-2016

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