Empty non-convex and convex four-gons in random point sets

Abstract

Let S be a set of n points distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points of S in its interior. We show that the expected number of empty non-convex four-gons with vertices from S is 12n 2 log n + o(n 2 log n) and the expected number of empty convex four-gons with vertices from S is Θ(n 2 ).

Holes and islands in random point sets

“…In this paper, we study the number of k-holes in random point sets in R 𝑑 . In particular, we obtain results that imply quadratic upper bounds on H k (n) for any fixed k and that both strengthen and generalize the bounds by Bárány and Füredi [4], Valtr [20], and Fabila-Monroy, Huemer, and Mitsche [10].…”

“…for any d and K. In the plane, Bárány and Füredi [4] proved EH K 2,3 (n) ≤ 2n 2 + O(n log n) for every K. This bound was later slightly improved by Valtr [19], who showed EH K 2,3 (n) ≤ 4 n 2 for any K. In the other direction, every set of n points in R d in general position contains at least n−1 d (d + 1)-holes [4,13]. The expected number EH K 2,4 (n) of 4-holes in random sets of n points in the plane was considered by Fabila-Monroy, Huemer, and Mitsche [9], who showed…”

“…for any K. In the other direction, every set of n points in R 𝑑 in general position contains at least [4,14]. The expected number EH K 2,4 (n) of 4-holes in random sets of n points in the plane was considered by Fabila-Monroy, Huemer, and Mitsche [10], who showed…”

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

“…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).…”

“…Besides empty simplices, we also consider larger k-holes. The expected number EH K 2,4 (n) of 4-holes in random planar sets of n points was considered by Fabila-Monroy, Huemer, and Mitsche [FMHM15], who showed…”