Improved Bounds for Geometric Permutations

TL;DR

This paper improves the upper bound on the number of geometric permutations for disjoint convex sets in higher dimensions, reducing the previously known bound by a significant factor.

Contribution

The authors establish a tighter upper bound of O(n^{2d-3} log n) for geometric permutations in ^d, improving upon Wenger's longstanding bound.

Findings

New upper bound of O(n^{2d-3} log n) for geometric permutations

Significant reduction from previous bound of O(n^{2d-2}

Applicable for convex sets in dimensions d  3

Abstract

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