An isoperimetric inequality for antipodal subsets of the discrete cube

A family of subsets of $\{1,2,\dots ,n\}$ 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 $\{1,2,\dots ,n\}$. 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.