  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.
