Improved Bounds for Geometric Permutations

Abstract: We show that the number of geometric permutations of an arbitrary collection of $n$ pairwise disjoint convex sets in $\mathbb{R}<sup>d$, for $d\geq 3$, is $O(n^{2d-3}\log n)$, improving Wenger's 20 years old bound of $O(n<sup>{2d-2})$.