On the Average Complexity of the $k$-Level

Abstract

Let $\mathcal{L}$ be an arrangement of $n$ lines in the Euclidean plane. The \emph{$k$-level} of $\mathcal{L}$ consists of all vertices $v$ of the arrangement which have exactly $k$ lines of $\mathcal{L}$ 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 $\Theta ((k+1{)}^{d-1})$.