Central limit theorems for random boundary polytopes

The number of faces of the convex hull of n independent and identically distributed random points chosen on the boundary of a smooth convex body in Rd is investigated. In dimensions two and three the number of k-faces is known to be constant almost surely and in dimension four and higher the variance is known to be non-zero if k≥1. We show that it is of order n. This is complemented by a central limit theorem with a Berry-Esseen bound which is of optimal order n−1∕2. We derive similar results for the Poissonized model, where additionally the number of random points is Poisson distributed. As a main tool, we develop a representation of the number of faces as a sum of exponentially stabilizing score functions.