An isoperimetric inequality for Hamming balls and local expansion in hypercubes

TL;DR

This paper establishes a sharp vertex isoperimetric inequality for Hamming balls in high-dimensional hypercubes, revealing a local expansion phenomenon that advances understanding of combinatorial structures.

Contribution

It introduces a new isoperimetric inequality for Hamming balls and uncovers a local expansion property in hypercubes, with sharp bounds for comparable set sizes.

Findings

Proved a sharp vertex isoperimetric inequality for Hamming balls.

Identified a local expansion phenomenon in hypercubes.

Provided bounds that are tight up to a constant factor.

Abstract

We prove a vertex isoperimetric inequality for the $n$-dimensional Hamming ball $\mathcal{B}_n(R)$ of radius $R$. The isoperimetric inequality is sharp up to a constant factor for sets that are comparable to $\mathcal{B}_n(R)$ in size. A key step in the proof is a local expansion phenomenon in hypercubes.