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 2 k , a family consisting of the union of two antipodal ( k − 1 ) -dimensional subcubes has the smallest possible edge boundary. [ABSTRACT FROM AUTHOR]