An isoperimetric inequality for antipodal subsets of the discrete cube. (May 2018) - Ellis, David; Leader, Imre
Abstract: 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.