Conjectures for cutting pizza with Coxeter arrangements

TL;DR

This paper explores conjectures related to the sign of a specific geometric quantity associated with Coxeter arrangements, using Taylor expansions and subarrangements to analyze its properties.

Contribution

It introduces a method to express the pizza quantity for Coxeter arrangements in terms of subarrangements and derives the first non-zero term in its Taylor expansion.

Findings

Identifies the sign pattern of the pizza quantity for certain Coxeter arrangements.

Develops a novel approach using 2-structures to analyze geometric quantities.

Provides the first non-zero term in the multivariate Taylor expansion of the pizza quantity.

Abstract

We are interested in conjecturing the sign of the pizza quantity P(H,B(a,R)) for the irreducible Coxeter arrangements H of type A_n, where n=2,3 bmod 4, and type D_n, where n is odd. Our approach is to express the pizza quantity in terms of the pizza quantity of subarrangements known as 2-structures, and we obtain the first non-zero term in the multivariate Taylor expansion.