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Our main contribution is a concentration inequality for the symmetric volumedifference of a $ C^2 $ convex body with positive Gaussian curvature and acircumscribed random polytope with a restricted number of facets, for anyprobability measure on the boundary with a positive density function. We alsoshow 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 afactor of $o(n)$. In addition, we provide a remarkable open problem which isthe natural geometric generalization of the famous and fundamental "balls andbins" problem from probability. This problem is tightly connected to theoptimality of random polytopes in high dimensions.

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