Empty convex polytopes in random point sets

Abstract

Given a set P of points in Rd, a convex hole (alternatively, empty convex polytope) of P is a convex polytope with vertices in P, containing no points of P in its interior. Let R be a bounded convex region in Rd. We show that if P is a set of n random points chosen independently and uniformly over R, then the expected number of vertices of the largest hole of P is Θ(log n/(log log n)), regardless of the shape of R. This generalizes the analogous result proved for the case d = 2 by Balogh, González-Aguilar, and Salazar.