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 ≥ 2 and N ≥ 10 n , there exists a polytope Pn,N in R n with at most N facets that satisfieswhere Dn is the n-dimensional Euclidean unit ball. This result closes gaps from several 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 can approximate the n-dimensional Euclidean ball with respect to the aforementioned metrics., 2010 Mathematics Subject Classification: Primary 52A22; Secondary 60D05.

A Concentration Inequality for Random Polytopes, Dirichlet–Voronoi Tiling Numbers and the Geometric Balls and Bins Problem

“…Our next result is an application of [27] and provides an improvement to a result of Zador [52] on the asymptotic behavior of the Dirichlet-Voronoi tiling number in R n , denoted by div n−1 . These numbers have numerous definitions.…”

“…This result implies that the conditional variance is bounded above by C(K, µ)N −(1+ 4 n−1 ) f (N ) −1 . The proof of Theorem 1 also gives a concentration inequality for the volume of the arbitrarily positioned random polytopes that were defined in [27]; these polytopes give an optimal approximation to the Euclidean unit ball B n when µ = σ is the uniform measure on the sphere. Corollary 1.…”

“…Due to the difficulty of finding best-approximating polytopes (even when n = 3), random polytopes can and have been used to derive sharp estimates for the approximation of convex bodies. In fact, it turns out that, asymptotically, random polytopes are almost as good as best-approximating polytopes under the symmetric difference metric [8,22,27,28,29,37,48]. In our first theorem, we derive a sharp concentration inequality (up to logarithmic factors) for the volume difference of a C 2 convex body K with positive Gaussian curvature and a random circumscribed polytope with N facets.…”

“…Kur [27] showed that the polytopes P n,N of Corollary 1 satisfy △ s (P n,N , B n ) ≤ 4n|P n,N △B n |. This inequality and Corollary 1 together imply that for all sufficiently large N , the following "large deviation" inequality holds:…”

“…Remark 1. For convex bodies K and L in R n , the surface area deviation △ s (K, L) is defined by (see, e.g., [23,27])…”