  The edge isoperimetric inequality in the discrete cube specifies, for each
pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of
an $m$-element subset of $\{0,1\}^{n}$; the extremal families (up to
automorphisms of the discrete cube) are initial segments of the lexicographic
ordering on $\{0,1\}^n$. We show that for any $m$-element subset $\mathcal{F}
\subset \{0,1\}^n$ and any integer $l$, if the edge boundary of $\mathcal{F}$
has size at most $g_n(m)+l$, then there exists an extremal family $\mathcal{G}
\subset \{0,1\}^n$ such that $|\mathcal{F} \Delta \mathcal{G}| \leq Cl$, where
$C$ is an absolute constant. This is best-possible, up to the value of $C$. Our
result can be seen as a `stability' version of the edge isoperimetric
inequality in the discrete cube, and as a discrete analogue of the seminal
stability result of Fusco, Maggi and Pratelli concerning the isoperimetric
inequality in Euclidean space.
