Our main contribution is an almost sharp concentration inequality for the symmetric volume difference of a $ C^2 $ convex body with strictly positive Gaussian curvature and a circumscribed random polytope with a restricted number of facets, for any probability measure on the boundary with a strictly 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)$, and we provide an interesting observation and an open conjecture about random partial sphere "covering" related to results of Erdős, Few and Rogers. This conjecture is closely connected to the optimality of random polytopes in high dimensions.
Finally, as an application of all of our results, we derive a lower bound for the Mahler volume product of polytopes with a restricted number of vertices.
This is joint work with Gil Kur, MIT (previously at Weizmann Institute of Science).