@article{71a032976b9e46e6bd6a03585c48f41f,

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

abstract = "For (Formula presented.), let S be a set of points in (Formula presented.) in general position. A set I of k points from S is a k-island in S if the convex hull (Formula presented.) of I satisfies (Formula presented.). A k-island in S in convex position is a k-hole in S. For (Formula presented.) and a convex body (Formula presented.) of volume 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-holes in S is in (Formula presented.). Our estimate improves and generalizes all previous bounds. In particular, we estimate the expected number of empty simplices in S by (Formula presented.). This is tight in the plane up to a lower-order term. Our method gives an asymptotically tight upper bound (Formula presented.) even in the much more general setting, where we estimate the expected number of k-islands in S.",

keywords = "convex position, k-hole, k-island, random point set, stochastic geometry",

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

note = "Funding Information: We would like to thank to the referees for their careful reading and comments that helped to improve the presentation of our results. 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 Grants FE 340/12‐1 and SCHE 2214/1‐1. 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: Czech Science Foundation (GACR), 18‐19158S; Center for Foundations of Modern Computer Science (Charles University), UNCE/SCI/004; Charles University, PRIMUS/17/SCI/3; European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, 810115; DFG, FE 340/12‐1, SCHE 2214/1‐1 Funding information Publisher Copyright: {\textcopyright} 2021 Wiley Periodicals LLC.",

year = "2022",

month = may,

day = "1",

doi = "10.1002/rsa.21037",

language = "English",

volume = "60",

pages = "308--326",

journal = "Random Structures and Algorithms",

issn = "1042-9832",

publisher = "John Wiley and Sons Ltd",

number = "3",

}