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Conjectures for cutting pizza with Coxeter arrangements

Richard Ehrenborg

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.

Conjectures for cutting pizza with Coxeter arrangements

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.

Paper Structure

This paper contains 6 sections, 12 theorems, 40 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Coxeter arrangements, pseudo-root systems and $2$-structures
  3. Multivariate Taylor expansion of the pizza quantity
  4. Computational results for the leading term
  5. Type $D_{n}$ in odd dimensions
  6. Concluding remarks

Key Result

Theorem 1.1

Let $\mathcal{H}$ be a Coxeter arrangement on a finite-dimensional inner product space $V$ such that $|\mathcal{H}| \geq \dim(V)$. Assume that the ball $\mathbb{B}(a,R)$ contains the origin, that is, $R \geq \|a\|$.

Figures (1)

  • Figure 1: A maximal matching on the set $\{1,2, \ldots, 9\}$. Note that the number of crossings is $3$ and the isolated vertex is at $2$. Hence the sign of this matching is $(-1)^{2-1} \cdot (-1)^{3} = +1$.

Theorems & Definitions (25)

  • Theorem 1.1: Ehrenborg--Morel--Readdy EMR_pizza
  • Theorem 1.2: Mabry--Deiermann MaDe
  • Conjecture 1.3: Ehrenborg--Morel--Readdy EMR_pizza
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • ...and 15 more