An isoperimetric inequality for antipodal subsets of the discrete cube

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.