Vertex-isoperimetric stability in the hypercube

TL;DR

This paper proves a stability result for Harper's Theorem in the hypercube, showing that sets with nearly minimal neighborhoods are close to Hamming balls, thus strengthening understanding of isoperimetric properties.

Contribution

It establishes a stability version of Harper's Theorem, linking near-minimal neighborhoods to proximity to Hamming balls in the hypercube.

Findings

Sets with nearly minimal neighborhoods are close to Hamming balls.

Provides quantitative bounds on the closeness to Hamming balls.

Enhances understanding of isoperimetric stability in hypercubes.

Abstract

Harper's Theorem states that, in a hypercube, among all sets of a given fixed size the Hamming balls have minimal closed neighbourhoods. In this paper we prove a stability-like result for Harper's Theorem: if the closed neighbourhood of a set is close to minimal in the hypercube, then the set must be very close to a Hamming ball around some vertex.