Approximation of the Euclidean ball by polytopes

Monika Ludwig ¹   ; Carsten Schütt ²   ; Elisabeth Werner ³

¹ Institut für Diskrete Mathematik und Geometrie Technische Universität Wien Wiedner Hauptstraße 8-10/104 1040 Wien, Austria
² Mathematisches Seminar Christian Albrechts Universität D-24098 Kiel, Germany
³ Department of Mathematics Case Western Reserve University Cleveland, OH 44106, U.S.A. and Université de Lille 1 UFR de Mathématique 59655 Villeneuve d'Ascq, France

Studia Mathematica, Tome 173 (2006) no. 1, pp. 1-18

Résumé

There is a constant $c$ such that for every $n\in\mathbb N$, there is an $N_{n}$ so that for every $N\geq N_{n}$ there is a polytope $P_{}$ in $\mathbb R^{n}$ with $N$ vertices and $$ \mathop{\rm vol}\nolimits _{n}(B_{2}^{n}\mathbin{\triangle} P) \leq c \mathop{\rm vol}\nolimits _{n}(B_{2}^{n})N^{-\frac{2}{n-1}} $$ where $B_{2}^{n}$ denotes the Euclidean unit ball of dimension $n$.

Détail

DOI : 10.4064/sm173-1-1

Keywords: there constant every mathbb there every geq there polytope mathbb vertices mathop vol nolimits mathbin triangle leq mathop vol nolimits frac n where denotes euclidean unit ball dimension