On the structure of subsets of the discrete cube with small edge boundary

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 $$\mathcal{F}$ \subset {0,1}^n$andanyinteger$l$,iftheedgeboundaryof$$\mathcal{F}$$hassizeatmost$g_n(m)+l$,thenthereexistsanextremalfamily$$\mathcal{G}$ \subset {0,1}^n$suchthat$|$\mathcal{F}$ \Delta $\mathcal{G}$| \leq Cl$,where$C$isanabsoluteconstant\mathrm{.}Thisisbest-possible,uptothevalueof$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.