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 F ⊂ (0,1)ⁿ and any integer l, if the edge boundary of F has size at most g_n(m)+l, then there exists an extremal family G⊂(0,1)ⁿ such that |FΔG| ≤ 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 [15] for the isoperimetric inequality in Euclidean space.