Umbrellas and polytopal approximation of the euclidean ball

Abstract

There are two positive, absolute constants $c_1$ and $c_2$ so that the volume of the difference set of the $d$-dimensional Euclidean ball and an inscribed polytope with n vertices is larger than $$c_{2}d{n}^{-\frac{2}{d-1}}vo{l}_d(B_2^d)$$ for $n\ge (c_{1}d{)}^{\frac{d-1}{2}}$.