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.

中文翻译：

---

立方体中顶点等操作的稳定性

我们证明了Harper立方体顶点等距不等式的稳定性，表明顶点边界接近最小可能的立方体子集接近（广义）汉明球。此外，我们获得了球状集的局部稳定性结果，该结果给出了与球之间的距离的顶点边界的清晰估计，因此我们的稳定性结果实质上是紧密的（模非单调现象）。我们还给出了Kruskal–Katona定理的相似结果，并将其应用于极值组合法中其他一些结果的新稳定性版本。