Isoperimetry in integer lattices

The edge isoperimetric problem for a graph G is to determine, for each n, the minimum number of edges leaving any set of n vertices. In general this problem is NP-hard, but exact solutions are known in some special cases, for example when G is the usual integer lattice. We solve the edge isoperimetric problem asymptotically for every Cayley graph on Z^d. The near-optimal shapes that we exhibit are zonotopes generated by line segments corresponding to the generators of the Cayley graph.