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Edge-Isoperimetric Inequalities in Chamber Graphs of Hyperplane Arrangements

Tilen Marc

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 $|\partial S|=Ω(|S|^{2/3})$. 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 $O(n^2\log(n/\varepsilon))$.

Edge-Isoperimetric Inequalities in Chamber Graphs of Hyperplane 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 , with , has edge boundary at least , and we conjecture that convex sets minimize the edge boundary among all chamber sets of a fixed size. We verify this conjecture in dimension . Our main result is a three-dimensional asymptotic inequality for arbitrary subsets of chambers: for arrangements in general position, every set occupying at most a fixed proportion of the chambers satisfies . As a consequence, for an arrangement of hyperplanes in general position in , the lazy simple random walk on the chamber graph has -mixing time .

Paper Structure

This paper contains 8 sections, 13 theorems, 192 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Work
  4. Approach and main results
  5. Preliminaries and basic lemmas
  6. First results and a conjecture
  7. Low dimensions
  8. Concluding Remarks and Further Directions

Key Result

Lemma 2.4

Let $S$ be a subset of chambers of a hyperplane arrangement $\mathcal{H}$ in general position in $\mathbb{R}^d$, and let $e\in[n]$. Define Let $A\in{\rm supp}(S)$ with $e\notin A$. If $c\in{\rm conn}(T_S(A\cup\{e\}))$, then $c$ lies on the boundary of exactly one connected component of $T_{S^+}(A)$. $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Three lines in $\mathbb{R}^2$ forming one bounded and six unbounded chambers. $S$ consists of the triangle and two adjacent chambers, giving ${\rm supp}(S)=\{\emptyset,\{1\},\{2\}\}$.
  • Figure 2: Example polytopes.
  • Figure 3: An arrangement in general position with $|S|=8$ (shaded) chambers. The boundary $\Sigma_S=\partial T_S$ decomposes along each line $H_f$ into intervals $\mathcal{I}_f$ (green) and intersection points $X_f$ (orange). In this example $\sum_f|\mathcal{I}_f|=5$, $\sum_f|X_f|=4$, giving $|\partial S|=9$. Three components of the $1$-strata are covered by points, and the remaining component is covered by an interval. The four unbounded intervals remain for the 0-stratum.
  • Figure 4: Local configurations at an endpoint of a $1$--stratum component (Claim \ref{['claim:tau_endpoint_lb_R3']}). Filled vertices represent chambers in $S$; hollow vertices are chambers not in $S$. Red edges cross the boundary $\partial S$. The bottom face is the distinguished square (four chambers in one $H_c$--layer all belonging to $S$).
  • Figure 5: Local configurations in the $2$--stratum dichotomy (Claim \ref{['clm:local_S2_dichotomy_R3']}). Chambers $000$ and $100$ always belong to $S$ (one sector of $H_e$). The exceptional pattern $X_{\mathrm{ex}}$ is the only case where $\tau(p)$ can be less than $\frac{1}{2}$.

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6: Kruskal--Katona
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 35 more