Edge boundary in a graph (${\partial }_{e}A$)

The edge boundary of a vertex set $A$ consists of graph edges with exactly one endpoint in $A$. In a $d$-regular graph,where $e(A)$ counts edges with both endpoints in $A$.

For $A\subseteq \{0,1{\}}^n$,Equivalently, $A$ spans at most $\frac{1}{2}|A|{\log }_2|A|$ cube edges. Induction on the dimension and concavity of binary entropy prove the inequality.

The binary entropy function isIt is concave on $[0,1]$ and satisfies $H_2(x)\ge 2x$ for $0\le x\le 1/2$ by comparison with the chord from $(0,0)$ to $(1/2,1)$.