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

Our main contribution is a concentration inequality for the symmetric volume difference of a C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C^2 $$\end{document} convex body with positive Gaussian curvature and a circumscribed random polytope with a restricted number of facets, for any probability measure on the boundary with a positive density function. We also show that the Dirichlet–Voronoi tiling numbers satisfy divn-1=(2πe)-1(n+lnn)+O(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \text {div}_{n-1} = (2\pi e)^{-1}(n+\ln n) + O(1)$$\end{document}, which improves a classical result of Zador by a factor of o(n). In addition, we provide a remarkable open problem which is the natural geometric generalization of the famous and fundamental “balls and bins” problem from probability. This problem is tightly connected to the optimality of random polytopes in high dimensions. Finally, as an application of the aforementioned results, we derive a lower bound for the maximal Mahler volume product of polytopes with a restricted number of vertices or facets.