Let $Q_n^r$ be the graph with vertex set ${\{-1,1\}}^n$ in which two vertices are joined if their Hamming distance is at most r. The edge-isoperimetric problem for $Q_n^r$ is that: For every $(n,r,M)$ such that $1\le r\le n$ and $1\le M\le 2^n$, determine the minimum edge-boundary size of a subset of vertices of $Q_n^r$ with a given size M. In this paper, we apply two different approaches to prove bounds for this problem. The first approach is a linear programming approach and the second is probabilistic. Our bound derived by the first approach generalizes the tight bound for $M=2^{n-1}$ derived by Kahn, Kalai, and Linial in 1989. Moreover, our bound is also tight for $M=2^{n-2}$ and $r\le \frac{n}{2}-1$. Our bounds derived by the second approach are expressed in terms of the noise stability, and they are shown to be asymptotically tight as $n\to \infty$ when $r=2\lfloor \frac{\beta n}{2}\rfloor +1$ and $M=\lfloor \alpha 2^n\rfloor$ for fixed $\alpha ,\beta \in (0,1)$, and is tight up to a factor 2 when $r=2\lfloor \frac{\beta n}{2}\rfloor$ and $M=\lfloor \alpha 2^n\rfloor$. In fact, the edge-isoperimetric problem is equivalent to a ball-noise stability problem which is a variant of the traditional (i.i.d.-) noise stability problem. Our results can be interpreted as bounds for the ball-noise stability problem.

让 ${问}_n^r$ 是具有顶点集的图 ${\{-1,1\}}^n$如果两个顶点的汉明距离最多为r ，则它们连接在一起。边等周长问题${问}_n^r$ 就是：对于每个 $(n,r,米)$ 这样 $1\le r\le n$ 和 $1\le 米\le 2^n$，确定顶点子集的最小边边界大小 ${问}_n^r$给定大小M。在本文中，我们应用了两种不同的方法来证明这个问题的界限。第一种方法是线性规划方法，第二种方法是概率。我们通过第一种方法得出的界概括了紧界$米=2^{n-1}$ 由 Kahn、Kalai 和 Linial 在 1989 年推导出来的。此外，我们的界限也很紧 $米=2^{n-2}$ 和 $r\le \frac{n}{2}-1$. 我们通过第二种方法得出的界限用噪声稳定性表示，并且它们被证明是渐近紧的$n\to \infty$ 什么时候 $r=2\lfloor \frac{\beta n}{2}\rfloor +1$ 和 $米=\lfloor \alpha 2^n\rfloor$ 对于固定 $\alpha ,\beta \in (0,1)$，并且紧到因子 2 时 $r=2\lfloor \frac{\beta n}{2}\rfloor$ 和 $米=\lfloor \alpha 2^n\rfloor$. 事实上，边缘等周问题等价于球噪声稳定性问题，它是传统（iid-）噪声稳定性问题的变体。我们的结果可以解释为球噪声稳定性问题的界限。