On the structure of subsets of the discrete cube with small edge boundary

TL;DR

This paper establishes a stability version of the edge isoperimetric inequality in the discrete cube, showing that sets with nearly minimal edge boundary are close to extremal lexicographic families.

Contribution

It proves that sets with edge boundary close to the minimum are structurally close to extremal families, extending the classical inequality with a stability result.

Findings

Sets with near-minimal edge boundary are close to lexicographic extremal families.

The stability bound is tight up to a universal constant.

The result parallels stability results in Euclidean isoperimetric inequalities.

Abstract

The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of an $m$-element subset of $\{0,1\}^{n}$; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on $\{0,1\}^n$. We show that for any $m$-element subset $\mathcal{F} \subset \{0,1\}^n$ and any integer $l$, if the edge boundary of $\mathcal{F}$ has size at most $g_n(m)+l$, then there exists an extremal family $\mathcal{G} \subset \{0,1\}^n$ such that $|\mathcal{F} \Delta \mathcal{G}| \leq Cl$, where $C$ is an absolute constant. This is best-possible, up to the value of $C$. Our result can be seen as a `stability' version of the edge isoperimetric inequality in the discrete cube, and as a discrete analogue of the seminal stability result of Fusco, Maggi and Pratelli concerning the isoperimetric inequality in Euclidean space.