Approximation of the Euclidean ball by polytopes with a restricted number of facets
Volume 251 / 2020
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} $ in $ \mathbb R ^n $ with at most $ N $ facets that satisfies $$ \Delta _{v}(D_n,P_{n,N}):=\operatorname{vol} _n (D_n \triangle P_{n,N} )\leq CN^{- 2/(n-1) } \operatorname{vol} _n (D_n ) $$ and $$ \Delta _{s}(D_n,P_{n,N}):=\operatorname{vol} _{n-1} (\partial (D_n\cup P_{n,N} ) ) - \operatorname{vol} _{n-1} (\partial (D_n\cap P_{n,N} ) ) \leq 4CN^{- 2/(n-1) } \operatorname{vol} _{n-1} (\partial D_n ), $$ where $ D_n $ is the $ n$-dimensional Euclidean unit ball. This result closes gaps in some papers of Hoehner, Ludwig, Schütt 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.
