Tight bounds on the expected number of holes in random point sets

Abstract

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 d-dimensional volume, we study the expected number EH K d,k (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 EH K d,k (n) by showing that, for all fixed integers d ≥ 2 and k ≥ d + 1, the number EH K d,k (n) is at least Ω(n d ). For some small holes, we even determine the leading constant lim n→∞ n −d EH K d,k (n) exactly. We improve the currently best known lower bound on lim n→∞ n −d EH K d,d+1 (n) by Reitzner and Temesvari (2019). In the plane, we show that the constant lim n→∞ n −2 EH K 2,k (n) is independent of K for every fixed k ≥ 3 and we compute it exactly for k = 4, improving earlier estimates by Fabila-Monroy, Huemer, and Mitsche (2015) and by the authors (2020).