An isoperimetric inequality for antipodal subsets of the discrete cube

We say a family of subsets of {1,2,…,n} is antipodal if it is closed under taking complements. We prove a best-possible isoperimetric inequality for antipodal families of subsets of {1,2,…,n} (of any size). Our inequality implies that for any k∈N, among all such families of size 2k, a family consisting of the union of two antipodal (k−1)-dimensional subcubes has the smallest possible edge boundary.