Abstract
A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of klevel vertices in an arrangement of n hyperplanes in R^d (vertices with exactly k of the hyperplanes passing below them). This is essentially a dual version of the kset problem, which, in a primal setting, seeks bounds for the maximum number of ksets determined by n points in R^d, where a kset is a subset of size k that can be separated from its complement by a hyperplane. The kset problem is still wide open even in the plane. In three dimensions, the best known upper and lower bounds are, respectively, O(nk^{3/2}) [M. Sharir et al., 2001] and nk * 2^{Omega(sqrt{log k})} [G. Tóth, 2000].
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 [M. Sharir and J. Zahl, 2017; H. Tamaki and T. Tokuyama, 2003], but the known upper bounds are rather weak for more general surfaces, already in three dimensions, except for the case of triangles [P. K. Agarwal et al., 1998]. The best known general bound, due to Chan [T. M. Chan, 2012] is O(n^{2.997}), for families of surfaces that satisfy certain (fairly weak) properties.
In this paper we consider the case of pseudoplanes in R^3 (defined in detail in the introduction), and establish the upper bound O(nk^{5/3}) for the number of klevel 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 kset problem, such as the Lovász Lemma and the Crossing Lemma.
BibTeX  Entry
@InProceedings{sharir_et_al:LIPIcs:2019:10466,
author = {Micha Sharir and Chen Ziv},
title = {{On the Complexity of the kLevel in Arrangements of Pseudoplanes}},
booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)},
pages = {62:162:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771047},
ISSN = {18688969},
year = {2019},
volume = {129},
editor = {Gill Barequet and Yusu Wang},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10466},
URN = {urn:nbn:de:0030drops104662},
doi = {10.4230/LIPIcs.SoCG.2019.62},
annote = {Keywords: klevel, pseudoplanes, arrangements, three dimensions, ksets}
}
Keywords: 

klevel, pseudoplanes, arrangements, three dimensions, ksets 
Collection: 

35th International Symposium on Computational Geometry (SoCG 2019) 
Issue Date: 

2019 
Date of publication: 

11.06.2019 