Tight bounds on the expected number of holes in random point sets
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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_d,k^K(n) is at least Ω(n^d). For some small holes, we even determine the leading constant lim_n →∞n^-dEH^K_d,k(n) exactly. We improve the currently best known lower bound on lim_n →∞n^-dEH^K_d,d+1(n) by Reitzner and Temesvari (2019). In the plane, we show that the constant lim_n →∞n^-2EH^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).
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