  For a general family of graphs on $\mathbb{Z}^n$, we translate the
edge-isoperimetric problem into a continuous isoperimetric problem in
$\mathbb{R}^n$. We then solve the continuous isoperimetric problem using the
Brunn-Minkowski inequality and Minkowski's theorem on Mixed Volumes. This
translation allows us to conclude, under a reasonable assumption about the
discrete problem, that the shapes of the optimal sets in the discrete problem
approach the shape of the optimal set in the continuous problem as the size of
the set grows. The solution is the zonotope defined as the Minkowski sum of the
edges of the original graph.
  We demonstrate the efficacy of this method by revisiting some previously
solved classical edge-isoperimetric problems. We then apply our method to some
discrete isoperimetric problems which had not previously been solved. The
complexity of those solutions suggest that it would be quite difficult to find
them using discrete methods only.
