 Equality cases of entropy cube edge-isoperimetry

For a nonempty subset of a hypercube graph, split it into sections of sizes $a,b$. The proof of the entropy edge bound uses $H_2(t)\ge 2t$, $t=\min (a,b)/(a+b)$. Strict concavity makes equality possible only at $t=0$ or $t=1/2$. In the first case one section is empty; in the second the sections coincide, since all possible crossing edges must be present. Induction forces a coordinate subcube. Conversely every coordinate subcube has the displayed edge count. At size $2^{n-1}$ these are exactly the two halves determined by fixing one coordinate.