On the complexity of the k-level in arrangements of pseudoplanes

Micha Sharir, Chen Ziv

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

Abstract

A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in Rd (vertices with exactly k of the hyperplanes passing below them). This is essentially a dual version of the k-set problem, which, in a primal setting, seeks bounds for the maximum number of k-sets determined by n points in Rd, where a k-set is a subset of size k that can be separated from its complement by a hyperplane. The k-set problem is still wide open even in the plane. In three dimensions, the best known upper and lower bounds are, respectively, O(nk3/2) [15] and nk · 2Ω(log k) [19]. In its dual version, the problem can be generalized by replacing hyperplanes by other families of surfaces (or curves in the planes). Reasonably sharp bounds have been obtained for curves in the plane [16, 18], but the known upper bounds are rather weak for more general surfaces, already in three dimensions, except for the case of triangles [1]. The best known general bound, due to Chan [7] is O(n2.997), for families of surfaces that satisfy certain (fairly weak) properties. In this paper we consider the case of pseudoplanes in R3 (defined in detail in the introduction), and establish the upper bound O(nk5/3) for the number of k-level vertices in an arrangement of n pseudoplanes. The bound is obtained by establishing suitable (and nontrivial) extensions of dual versions of classical tools that have been used in studying the primal k-set problem, such as the Lovász Lemma and the Crossing Lemma.

Original languageEnglish
Title of host publication35th International Symposium on Computational Geometry, SoCG 2019
EditorsGill Barequet, Yusu Wang
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771047
DOIs
StatePublished - 1 Jun 2019
Event35th International Symposium on Computational Geometry, SoCG 2019 - Portland, United States
Duration: 18 Jun 201921 Jun 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume129
ISSN (Print)1868-8969

Conference

Conference35th International Symposium on Computational Geometry, SoCG 2019
Country/TerritoryUnited States
CityPortland
Period18/06/1921/06/19

Keywords

  • Arrangements
  • K-level
  • K-sets
  • Pseudoplanes
  • Three dimensions

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