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DOI: 10.4230/LIPIcs.SoCG.2022.57
URN: urn:nbn:de:0030-drops-160652
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16065/
Patáková, Zuzana ; Sharir, Micha
Covering Points by Hyperplanes and Related Problems
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Abstract
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.
BibTeX - Entry
@InProceedings{patakova_et_al:LIPIcs.SoCG.2022.57,
author = {Pat\'{a}kov\'{a}, Zuzana and Sharir, Micha},
title = {{Covering Points by Hyperplanes and Related Problems}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {57:1--57:7},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16065},
URN = {urn:nbn:de:0030-drops-160652},
doi = {10.4230/LIPIcs.SoCG.2022.57},
annote = {Keywords: Rich hyperplanes, Incidences, Covering points by hyperplanes}
}
| Keywords: | Rich hyperplanes, Incidences, Covering points by hyperplanes | |
| Seminar: | 38th International Symposium on Computational Geometry (SoCG 2022) | |
| Issue date: | 2022 | |
| Date of publication: | 01.06.2022 |
