@inbook{fda5c30d61bc4303b23c9eb1bb2dcfd0,

title = "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 Rd 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⊆ Rd of unit volume, we study the expected number EHd,kK(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 EHd,kK(n)≥Ω(nd) for all fixed d≥ 2 and k≥ d+ 1. For small holes, we even determine the leading constant limn→∞n-dEHd,kK(n) exactly. We improve the best known lower bound on limn→∞n-dEHd,d+1K(n) and we show that our bound is tight for d≤ 3. We show that limn→∞n-2EH2,kK(n) is independent of K for every fixed k≥ 3 and we compute it exactly for k= 4, improving several earlier estimates.",

keywords = "Convex position, Holes, Random point set, Stochastic geometry",

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

note = "Funding Information: Keywords: Stochastic geometry · Random point set · Convex position · Holes M. Balko—was supported by the grant no. 21-32817S of the Czech Science Foundation (GACˇR), 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). M. Scheucher—was partially supported by DFG Grant FE 340/12-1 and by the internal research funding “Post-Doc-Funding” from Technische Universit{\"a}t Berlin. We also gratefully acknowledge support from the internal research program IFFP 2016–2020 by the FernUniversi{\"a}t in Hagen. P. Valtr—was supported by the grant no. 21-32817S of the Czech Science Foundation (GACˇR) and by the PRIMUS/17/SCI/3 project of Charles University. Funding Information: M. Balko?was supported by the grant no. 21-32817S of the Czech Science Foundation (GA?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). M. Scheucher?was partially supported by DFG Grant FE 340/12-1 and by the internal research funding ?Post-Doc-Funding? from Technische Universit?t Berlin. We also gratefully acknowledge support from the internal research program IFFP 2016?2020 by the FernUniversi?t in Hagen. P. Valtr?was supported by the grant no. 21-32817S of the Czech Science Foundation (GA?R) and by the PRIMUS/17/SCI/3 project of Charles University. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.",

year = "2021",

month = jan,

day = "1",

doi = "10.1007/978-3-030-83823-2_64",

language = "English",

series = "Trends in Mathematics",

publisher = "Springer Science and Business Media Deutschland GmbH",

pages = "411--416",

booktitle = "Trends in Mathematics",

address = "Germany",

}