Isoperimetric stability in lattices

We obtain isoperimetric stability theorems for general Cayley digraphs on $\bZ^d$. For any fixed $B$ that generates $\bZ^d$ over $\bZ$, we characterise the approximate structure of large sets $A$ that are approximately isoperimetric in the Cayley digraph of $B$: we show that $A$ must be close to a set of the form $kZ \cap \bZ^d$, where for the vertex boundary $Z$ is the conical hull of $B$, and for the edge boundary $Z$ is the zonotope generated by $B$.