  We prove a stability version of Harper's cube vertex isoperimetric
inequality, showing that subsets of the cube with vertex boundary close to the
minimum possible are close to (generalised) Hamming balls. Furthermore, we
obtain a local stability result for ball-like sets that gives a sharp estimate
for the vertex boundary in terms of the distance from a ball, and so our
stability result is essentially tight (modulo a non-monotonicity phenomenon).
We also give similar results for the Kruskal--Katona Theorem and applications
to new stability versions of some other results in Extremal Combinatorics.
