We obtain isoperimetric stability theorems for general Cayley digraphs on $\mathbb{ Z}^d$. For any fixed $B$ that generates $\mathbb{ Z}^d$ over $\mathbb{ Z}$, 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 $kZ \cap \mathbb{ Z}^d$, where for the vertex boundary $Z$ is the conical hull of $B$, and for the edge boundary $Z$ is the zonotope generated by $B$.
Categories: math.CO