Covering Points by Hyperplanes and Related Problems

Zuzana Patáková, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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.

Original languageEnglish
Title of host publication38th International Symposium on Computational Geometry, SoCG 2022
EditorsXavier Goaoc, Michael Kerber
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772273
StatePublished - 1 Jun 2022
Event38th International Symposium on Computational Geometry, SoCG 2022 - Berlin, Germany
Duration: 7 Jun 202210 Jun 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference38th International Symposium on Computational Geometry, SoCG 2022


  • Covering points by hyperplanes
  • Incidences
  • Rich hyperplanes

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