@inproceedings{de8a39158cb24c468a7e5ef350c3503e,

title = "Covering Points by Hyperplanes and Related Problems",

abstract = "For a set P of n points in Rd, for any d = 2, a hyperplane h is called k-rich with respect to P if it contains at least k points of P. Answering and generalizing a question asked by Peyman Afshani, we show that if the number of k-rich hyperplanes in Rd, d = 3, is at least ?(nd/ka + n/k), with a sufficiently large constant of proportionality and with d = a < 2d- 1, then there exists a (d- 2)-flat that contains ?(k(2d-1-a)/(d-1)) points of P. We also present upper bound constructions that give instances in which the above lower bound is tight. An extension of our analysis yields similar lower bounds for k-rich spheres.",

keywords = "Covering points by hyperplanes, Incidences, Rich hyperplanes",

author = "Zuzana Pat{\'a}kov{\'a} and Micha Sharir",

note = "Publisher Copyright: {\textcopyright} Zuzana Patkov and Micha Sharir; licensed under Creative Commons License CC-BY 4.0; null ; Conference date: 07-06-2022 Through 10-06-2022",

year = "2022",

month = jun,

day = "1",

doi = "10.4230/LIPIcs.SoCG.2022.57",

language = "אנגלית",

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

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

editor = "Xavier Goaoc and Michael Kerber",

booktitle = "38th International Symposium on Computational Geometry, SoCG 2022",

}