= Equality cases of entropy cube edge-isoperimetry
{title2=$e(A)=\tfrac12|A|\log_2|A|\iff A\text{ is a coordinate subcube}$}
For a nonempty subset of a <hypercube graph>, split it into sections of sizes $a,b$. The proof of the entropy edge bound uses $H_2(t)\ge2t$, $t=\min(a,b)/(a+b)$. 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.
Back to article page
