Tight Bounds on the Expected Number of Holes in Random Point Sets

For integers d≥ 2 and k≥ d+ 1, a k-hole in a set S of points in general position in R^d is a k-tuple of points from S in convex position such that the interior of their convex hull does not contain any point from S. For a convex body K⊆ R^d of unit volume, we study the expected number EHd,kK(n) of k-holes in a set of n points drawn uniformly and independently at random from K. We prove an asymptotically tight lower bound on EHd,kK(n)≥Ω(nd) for all fixed d≥ 2 and k≥ d+ 1. For small holes, we even determine the leading constant limn→∞n-dEHd,kK(n) exactly. We improve the best known lower bound on limn→∞n-dEHd,d+1K(n) and we show that our bound is tight for d≤ 3. We show that limn→∞n-2EH2,kK(n) is independent of K for every fixed k≥ 3 and we compute it exactly for k= 4, improving several earlier estimates.