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Covering Points by Hyperplanes and Related Problems

Authors Zuzana Patáková , Micha Sharir

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Author Details

Zuzana Patáková
  • Department of Algebra, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Micha Sharir
  • School of Computer Science, Tel Aviv University, Tel Aviv, Israel


The authors thank Peyman Afshani for sharing his thoughts with us concerning this problem.

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Zuzana Patáková and Micha Sharir. Covering Points by Hyperplanes and Related Problems. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 57:1-57:7, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


For a set P of n points in ℝ^d, 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 ℝ^d, d ≥ 3, is at least Ω(n^d/k^α + n/k), with a sufficiently large constant of proportionality and with d ≤ α < 2d-1, then there exists a (d-2)-flat that contains Ω(k^{(2d-1-α)/(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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Rich hyperplanes
  • Incidences
  • Covering points by hyperplanes


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