Hyperplane Arrangements with Large Average Diameter

TL;DR

This paper investigates the maximum average diameter of bounded cells in simple hyperplane arrangements, proving conjectures in low dimensions and providing exact values and bounds in specific cases.

Contribution

It proves the conjecture in dimension 2, determines exact maximum average diameters for certain arrangements, and provides asymptotically tight bounds in fixed dimensions.

Findings

Conjecture holds in dimension 2.

Exact maximum average diameter for arrangements with up to dimension+2 hyperplanes in dimension 2.

Bounds in dimension 3 are asymptotically equal to the dimension.

Abstract

The largest possible average diameter of a bounded cell of a simple hyperplane arrangement is conjectured to be not greater than the dimension. We prove that this conjecture holds in dimension 2, and is asymptotically tight in fixed dimension. We give the exact value of the largest possible average diameter for all simple arrangements in dimension 2, for arrangements having at most the dimension plus 2 hyperplanes, and for arrangements having 6 hyperplanes in dimension 3. In dimension 3, we give lower and upper bounds which are both asymptotically equal to the dimension.