Isoperimetric stability in lattices

TL;DR

This paper establishes stability results for isoperimetric inequalities in Cayley digraphs on integer lattices, characterizing large near-minimal boundary sets as close to structured geometric objects.

Contribution

It provides the first stability theorems for isoperimetric problems in Cayley digraphs on $\

Findings

Large sets with near-minimal boundary are close to lattice slabs or zonotopes.

Characterization of approximate isoperimetric sets in Cayley digraphs on $\

Abstract

We obtain isoperimetric stability theorems for general Cayley digraphs on $\mathbb{Z}^d$. For any fixed $B$ that generates $\mathbb{Z}^d$ over $\mathbb{Z}$, we characterise the approximate structure of large sets $A$ that are approximately isoperimetric in the Cayley digraph of $B$: we show that $A$ must be close to a set of the form $kZ \cap \mathbb{Z}^d$, where for the vertex boundary $Z$ is the conical hull of $B$, and for the edge boundary $Z$ is the zonotope generated by $B$.