Approximating a convex body by a polytope using the epsilon-net theorem

Abstract

Giving a joint generalization of a result of Brazitikos, Chasapis and Hioni and results of Giannopoulos and Milman, we prove that roughly $\lceil \frac{d}{(1-\vartheta {)}^d}\ln \frac{1}{(1-\vartheta {)}^d}\rceil$ points chosen uniformly and independently from a centered convex body $K$ in ${\mathbb{R}}^d$ yield a polytope $P$ for which $\vartheta K\subseteq P\subseteq K$ holds with large probability. The proof is simple, and relies on a combinatorial tool, the $\epsilon$-net theorem.

4 pages