Definition (Edge-boundary). Let $G$ be a graph and let $A\subset V(G)$. The edge-boundary $\partial A$ of $A$ is the set of edges $xy$ such that $y\notin A$.

If $G={\mathbb{Z}}^n$ or ${\{0,1\}}^n$ and $i\in [n]$, then the $i$-th boundary ${\partial }_{i}A$ is the set of edges $xy\in \partial A$ such that $x-y=\pm e_i$, i.e. ${\partial }_{i}A$ consists of deges pointing in direction $i$.