  We consider the family of graphs whose vertex set is $\mathbb{Z}^n$ where two
vertices are connected by an edge when their $\ell_\infty$-distance is 1.
Towards an edge isoperimetric inequality for this graph, we calculate the edge
boundary of any finite set $S \subset \mathbb{Z}^n$. This boundary calculation
leads to a desire to show that a set with optimal edge boundary has no ``gaps''
in any direction $\epsilon \in \{-1,0,1\}^n, \epsilon \not=0$. We show that one
can find a set with optimal edge boundary that does not have gaps in any
direction $e_i$ (or $-e_i$) where $e_i$ is the standard basis vector.
