Edge-isoperimetric theorem for binary initial segments

Among subsets of the $n$-dimensional hypercube graph with $a$ vertices, initial segments of the binary order on the discrete cube minimize the edge boundary. Its exact minimum is $na-2{\sum }_{j=0}^{a-1}{s}_2(j)$. This strengthens the entropy lower bound on the same boundary.

Among families of a given size in the Boolean hypercube, a binary initial segment maximizes the number of contained two-dimensional cube faces. Under section compression in binary order, faces crossing the selected coordinate correspond to internal edges in the intersection of its two sections. The edge extremal theorem and mathematical induction control that contribution and the faces within the sections.