Edge-Isoperimetric Inequalities in Chamber Graphs of Hyperplane Arrangements

TL;DR

This paper investigates edge-isoperimetric inequalities in chamber graphs of hyperplane arrangements using topological methods, establishing bounds, conjecturing minimization properties, and analyzing random walk mixing times.

Contribution

It introduces topological techniques to derive bounds on edge boundaries, proposes a conjecture on convex sets minimizing boundary size, and provides asymptotic inequalities for random walk mixing times.

Findings

Convex chamber sets of certain sizes have explicit lower bounds on edge boundary.

Verified the convex boundary minimization conjecture in 2D.

Established an $Ω(|S|^{2/3})$ boundary size for subsets in 3D arrangements.

Abstract

We study edge-isoperimetric inequalities in chamber graphs of affine hyperplane arrangements. Our approach is topological: to a set of chambers we associate its thickening in Euclidean space and estimate its edge boundary through the induced stratification by intersections of arrangement hyperplanes. This yields general lower bounds for a broad class of sets. We show that a convex set of chambers of size $\sum_{i=0}^d \binom{k}{i}$, with $k\ge d-1$, has edge boundary at least $\sum_{i=0}^{d-1}\binom{k}{i}$, and we conjecture that convex sets minimize the edge boundary among all chamber sets of a fixed size. We verify this conjecture in dimension $2$. Our main result is a three-dimensional asymptotic inequality for arbitrary subsets of chambers: for arrangements in general position, every set $S$ occupying at most a fixed proportion of the chambers satisfies $|\partial S|=\Omega(|S|^{2/3})$. As a consequence, for an arrangement of $n$ hyperplanes in general position in $\mathbb R^3$, the lazy simple random walk on the chamber graph has $\varepsilon$-mixing time $O(n^2\log(n/\varepsilon))$.