On the Average Complexity of the $k$-Level

• Main Authors: Chiu, Man-Kwun, Felsner, Stefan, Scheucher, Manfred, Schnider, Patrick, Steiner, Raphael, Valtr, Pavel
• Format: Article in Journal/Newspaper
• Language: unknown
• Published: arXiv 2019
• Subjects: Computational Geometry cs.CG; Combinatorics math.CO; FOS Computer and information sciences; FOS Mathematics; G.2.1; I.3.5; 52C30, 68Q25; South Pole; South pole
• Online Access: https://dx.doi.org/10.48550/arxiv.1911.02408; https://arxiv.org/abs/1911.02408

Let ${\cal L}$ be an arrangement of $n$ lines in the Euclidean plane. The \emph{$k$-level} of ${\cal L}$ consists of all vertices $v$ of the arrangement which have exactly $k$ lines of ${\cal L}$ passing below $v$. The complexity (the maximum size) of the $k$-level in a line arrangement has been widely studied. In 1998 Dey proved an upper bound of $O(n\cdot (k+1)^{1/3})$. Due to the correspondence between lines in the plane and great-circles on the sphere, the asymptotic bounds carry over to arrangements of great-circles on the sphere, where the $k$-level denotes the vertices at distance at most $k$ to a marked cell, the \emph{south pole}. We prove an upper bound of $O((k+1)^2)$ on the expected complexity of the $k$-level in great-circle arrangements if the south pole is chosen uniformly at random among all cells. We also consider arrangements of great $(d-1)$-spheres on the sphere $\mathbb{S}^d$ which are orthogonal to a set of random points on $\mathbb{S}^d$. In this model, we prove that the expected complexity of the $k$-level is of order $Θ((k+1)^{d-1})$.