Central Limit Theorem for the Volume of Random Polytopes with Vertices on the Boundary
Given a convex body K with smooth boundary \(\partial K\), select a fixed number n of uniformly distributed random points from \(\partial K\). The convex hull \(K_n\) of these points is a random polytope having all its vertices on the boundary of K. The closeness of the volume of \(K_n\) to a Gaussian random variable is investigated in terms of the Kolmogorov distance by combining a version of Stein’s method with geometric estimates for the surface body of K.
KeywordsCentral limit theorem Random polytope Surface body Stochastic geometry
Mathematics Subject Classification52A22 60D05 60F05
I am grateful to Julian Grote (Bochum) for stimulating discussions and helpful comments. I also thank two anonymous referees for careful reading and their suggestions.
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