An isoperimetric inequality for antipodal subsets of the discrete cube

TL;DR

This paper establishes an optimal isoperimetric inequality for antipodal subset families of the discrete cube, identifying the minimal edge boundary configurations among such families.

Contribution

It proves the best-possible isoperimetric inequality for antipodal families of subsets of the discrete cube, characterizing minimal boundary structures.

Findings

Antipodal families of size 2^k have minimal boundary when they are unions of a (k-1)-dimensional subcube and its antipode.

The inequality is tight and characterizes extremal families.

The result generalizes classical isoperimetric inequalities to antipodal set families.

Abstract

A family of subsets of $\{1,2,\ldots,n\}$ is said to be {\em antipodal} if it is closed under taking complements. We prove a best-possible isoperimetric inequality for antipodal families of subsets of $\{1,2,\ldots,n\}$. Our inequality implies that for any $k \in \mathbb{N}$, among all such families of size $2^k$, a family consisting of the union of a $(k-1)$-dimensional subcube and its antipode has the smallest possible edge boundary.