Approximation of the Euclidean ball by polytopes with a restricted number of facets

Abstract: We prove that there is an absolute constant $ C$ such that for every $ n \geq 2 $ and $ N\geq 10^n, $ there exists a polytope $ P_{n,N} \subset \mathbb{R}^n $ with at most $ N $ facets that satisfies $$\Delta_{v}(D_n,P_{n,N}):=\text{vol}n\left(D_n \Delta P{n,N}\right)\leq Cn^{-2/(n-1}\text{vol}_n\left(D_n\right)$$ and $$ \Delta_{s}(D_n,P_{n,N}):=\text{vol}{n-1}\left(\partial\left(D_n\cup P{n,N}\right)\right) - \text{vol}{n-1}\left(\partial\left(D_n\cap P{n,N}\right)\right) \leq 4CN^{-\frac{2}{n-1}} \text{vol}_{n-1}\left(\partial D_n\right), $$ where $ D_n $ is the $ n$-dimensional Euclidean unit ball. This result closes gaps from several papers of Hoehner, Ludwig, Sch\"utt and Werner. The upper bounds are optimal up to absolute constants. This result shows that a polytope with an exponential number of facets (in the dimension) can approximate the $ n$-dimensional Euclidean ball with respect to the aforementioned distances.