  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 $B$.
