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We prove that roughly \(\frac{d}{(1-\vartheta )^d}\ln \frac{1}{(1-\vartheta )^d}\) 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. This gives a joint generalization of results of Brazitikos, Chasapis and Hioni and of Giannopoulos and Milman.

Approximating a Convex Body by a Polytope Using the Epsilon-Net Theorem