Improved Bounds for Geometric Permutations

Natan Rubin, Haim Kaplan, Micha Sharir

Abstract: We show that the number of geometric permutations of an arbitrary collection of n pairwise disjoint convex sets in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> , for d ≥ 3, is O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2d-3</sup> log n), improving Wenger's 20 years old bound of O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2d-2</sup> ).