  We consider the family of graphs whose vertex set is Z^k where two vertices
are connected by an edge when their l\infty-distance is 1. We prove the optimal
vertex isoperimetric inequality for this family of graphs. That is, given a
positive integer n, we find a set A \subset Z^k of size n such that the number
of vertices who share an edge with some vertex in A is minimized. These sets of
minimal boundary are nested, and the proof uses the technique of compression.
  We also show a method of calculating the vertex boundary for certain subsets
in this family of graphs. This calculation and the isoperimetric inequality
allow us to indirectly find the sets which minimize the function calculating
the boundary.
