On the Average Complexity of the k-Level
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Let A be an arrangement of n lines in the Euclidean plane. The <i>k-level<i> of A consists of all intersection points v of lines in A which have exactly k lines of A passing below v. The complexity of the k-level in a line arrangement has been widely studied. In 1998 Dey proved an upper bound of O(n· (k+1)^1/3). We investigate the complexity of k-levels in random line and hyperplane arrangements. When the arrangement is obtained from any fixed projective line arrangement of n lines by choosing a random cell to contain the south-pole, we prove an upper bound of O((k+1)^2) on the expected complexity of the k-level. As a byproduct we show that the complexity of any (≤ j)-zone in a d-dimensional simple arrangement of n hyperplanes is of order Θ((j+1)n^d-1). The classical zone theorem is the case j=0. We also consider arrangements of great (d-1)-spheres on the sphere S^d which are orthogonal to a set of random points on S^d. In this model we prove that the expected complexity of the k-level is of order Θ((k+1)^d-1).
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