Fine approximation of convex bodies by polytopes

Abstract

We prove that for every convex body $K$ with the center of mass at the origin and every $\epsilon \in \left(0,\frac{1}{2}\right)$, there exists a convex polytope $P$ with at most $e^{O(d)}{\epsilon }^{-\frac{d-1}{2}}$ vertices such that $(1-\epsilon )K\subset P\subset K$.

12 pages, 5 figures