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

Keywords: denote mathcal family convex bodies minimal width mathcal ask what greatest number varlambda mathcal every mathcal contains polytope vertices which varlambda mathcal leq lower estimate varlambda mathcal geq based estimates smallest radius lfloor rfloor antipodal pairs spherical caps cover unit sphere varlambda mathcal geq frac sqrt varlambda mathcal geq cos frac lfloor rfloor every geq consider dual question estimating smallest number delta mathcal every mathcal there exists polytope supset facets which diam diam leq delta mathcal upper bound delta mathcal geq particular delta mathcal leq cos frac lfloor rfloor geq