Projection volumes of hyperplane arrangements

TL;DR

This paper proves a geometric-combinatorial relation between average projection volumes of hyperplane arrangement cones and the characteristic polynomial, confirming a conjecture for all finite real reflection arrangements.

Contribution

It establishes a universal formula linking projection volumes to the characteristic polynomial for all finite real hyperplane arrangements, extending previous conjectures.

Findings

Average projection volumes equal the coefficients of the characteristic polynomial.

Confirmed the conjecture for all finite real reflection arrangements.

Connected angle sums of zonotopes to the characteristic polynomial of the intersection lattice.

Abstract

We prove that for any finite real hyperplane arrangement the average projection volumes of the maximal cones is given by the coefficients of the characteristic polynomial of the arrangement. This settles the conjecture of Drton and Klivans that this held for all finite real reflection arrangements. The methods used are geometric and combinatorial. As a consequence we determine that the angle sums of a zonotope are given by the characteristic polynomial of the order dual of the intersection lattice of the arrangement.