Vertex-isoperimetric inequality in the discrete cube ($|N(\mathcal{A})|\ge |N(I_{|\mathcal{A}|})|$)

For the Boolean hypercube, the initial segment of the simplicial order on the discrete cube minimizes the size of the closed vertex neighbourhood among families of a given size. Equivalently, it minimizes the external vertex boundary. This is a vertex assertion, distinct from the edge-isoperimetric inequality in the discrete cube.

Adjoin all lower levels to a uniform set family before applying Harper theorem. Its remaining neighbourhood contribution is the upper shadow, so lexicographic initial segments minimize upper shadows. Complementation and reversal of coordinates turn this into colexicographic order minimizing the lower shadow, giving the Kruskal-Katona theorem.