On the Complexity of the k-Level in Arrangements of Pseudoplanes
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 R^d (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 R^d, 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(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 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.
k-level
pseudoplanes
arrangements
three dimensions
k-sets
Mathematics of computing~Combinatorics
Theory of computation~Computational geometry
62:1-62:15
Regular Paper
Work on this paper was supported by Grants 892/13 and 260/18 from the Israel Science Foundation.
A full version of this paper is available at https://arxiv.org/abs/1903.07196.
Micha
Sharir
Micha Sharir
School of Computer Science, Tel Aviv University, Tel Aviv, Israel
http://www.cs.tau.ac.il/~michas/
Work on this paper was also supported by Grant G-1367-407.6/2016 from the German-Israeli Foundation for Scientific Research and Development, by the Blavatnik Research Fund in Computer Science at Tel Aviv University, and by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11).
Chen
Ziv
Chen Ziv
School of Computer Science, Tel Aviv University, Tel Aviv, Israel
10.4230/LIPIcs.SoCG.2019.62
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http://arxiv.org/abs/1903.07196
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Micha Sharir and Chen Ziv
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