Stability for vertex isoperimetry in the cube

Understand

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.