@inproceedings{33b94cc5a73a46f79a541171416d8bfd,

title = "Holes and islands in random point sets",

abstract = "For d ∈ N, let S be a finite set of points in Rd 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 Rd 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(nd). 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 2d−1 · d! · (nd ). Our results improve and generalize previous bounds by B{\'a}r{\'a}ny and F{\"u}redi [4], Valtr [19], Fabila-Monroy and Huemer [8], and Fabila-Monroy, Huemer, and Mitsche [9].",

keywords = "Convex position, Empty polytope, Erd{\H o}s-Szekeres type problem, Horton set, K-hole, K-island, Random point set, Stochastic geometry",

author = "Martin Balko and Manfred Scheucher and Pavel Valtr",

note = "Funding Information: Martin Balko: was supported by the grant no. 18-19158S of the Czech Science Foundation (GA{\v C}R), by the Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004), and by the PRIMUS/17/SCI/3 project of Charles University. This article is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 810115). Manfred Scheucher: was supported by DFG Grant FE 340/12-1. Pavel Valtr: was supported by the grant no. 18-19158S of the Czech Science Foundation (GA{\v C}R) and by the PRIMUS/17/SCI/3 project of Charles University. Funding Information: Funding Martin Balko: was supported by the grant no. 18-19158S of the Czech Science Foundation (GA{\v C}R), by the Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004), and by the PRIMUS/17/SCI/3 project of Charles University. This article is part of a project that has received funding from the European Research Council (ERC) under the European Union{\textquoteright}s Horizon 2020 research and innovation programme (grant agreement No 810115). Manfred Scheucher: was supported by DFG Grant FE 340/12-1. Pavel Valtr: was supported by the grant no. 18-19158S of the Czech Science Foundation (GA{\v C}R) and by the PRIMUS/17/SCI/3 project of Charles University. Publisher Copyright: {\textcopyright} Martin Balko, Manfred Scheucher, and Pavel Valtr; licensed under Creative Commons License CC-BY 36th International Symposium on Computational Geometry (SoCG 2020).; 36th International Symposium on Computational Geometry, SoCG 2020 ; Conference date: 23-06-2020 Through 26-06-2020",

year = "2020",

month = jun,

day = "1",

doi = "10.4230/LIPIcs.SoCG.2020.14",

language = "English",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

editor = "Sergio Cabello and Chen, {Danny Z.}",

booktitle = "36th International Symposium on Computational Geometry, SoCG 2020",

address = "Germany",

}