Holes and islands in random point sets
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For d ∈ℕ, let S be a finite set of points in ℝ^d in general position. A set H of k points from S is a k-hole in S if all points from H lie on the boundary of the convex hull conv(H) of H and the interior of conv(H) does not contain any point from S. A set I of k points from S is a k-island in S if conv (I) ∩ S = I. Note that each k-hole in S is a k-island in S. For fixed positive integers d, k and a convex body K in ℝ^d with d-dimensional Lebesgue measure 1, let S be a set of n points chosen uniformly and independently at random from K. We show that the expected number of k-islands in S is in O(n^d). In the case k=d+1, we prove that the expected number of empty simplices (that is, (d+1)-holes) in S is at most 2^d-1· d! ·nd. Our results improve and generalize previous bounds by Bárány and Füredi (1987), Valtr (1995), Fabila-Monroy and Huemer (2012), and Fabila-Monroy, Huemer, and Mitsche (2015).
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