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

TL;DR

This paper establishes a concentration inequality for random polytopes approximating convex bodies, improves bounds on Dirichlet-Voronoi tiling numbers, and introduces a new geometric generalization of the balls and bins problem.

Contribution

It provides a novel concentration inequality for symmetric volume differences and refines classical bounds on Dirichlet-Voronoi tiling numbers, linking them to high-dimensional probability.

Findings

Concentration inequality for symmetric volume difference of convex bodies and random polytopes.

Improved asymptotic bounds on Dirichlet-Voronoi tiling numbers.

Formulation of a new geometric problem related to the balls and bins paradigm.

Abstract

Our main contribution is a concentration inequality for the symmetric volume difference of a $ C^2 $ 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 $ \text{div}_{n-1} = (2\pi e)^{-1}(n+\ln n) + O(1)$, 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.