A stability result for the cube edge isoperimetric inequality

We prove the following stability version of the edge isoperimetric inequality for the cube: any subset of the cube with average boundary degree within $K$ of the minimum possible is $\varepsilon$-close to a union of $L$ disjoint cubes, where $L \leq L(K,\varepsilon )$ is independent of the dimension. This extends a stability result of Ellis, and can viewed as a dimension-free version of Friedgut's junta theorem.