In this paper we prove that the $\ell_0$ isoperimetric coefficient for any axis-aligned cubes, $\psi_{\mathcal{C}}$, is $\Theta(n^{-1/2})$ and that the isoperimetric coefficient for any measurable body $K$, $\psi_K$, is of order $O(n^{-1/2})$. As a corollary we deduce that axis-aligned cubes essentially "maximize" the $\ell_0$ isoperimetric coefficient: There exists a positive constant $q > 0$ such that $\psi_K \leq q \cdot \psi_{\mathcal{C}}$, whenever $\mathcal{C}$ is an axis-aligned cube and $K$ is any measurable set. Lastly, we give immediate applications of our results to the mixing time of Coordinate-Hit-and-Run for sampling points uniformly from convex bodies.

翻译：本文证明了对任意轴对齐立方体，其$\ell_0$等周系数$\psi_{\mathcal{C}}$为$\Theta(n^{-1/2})$，且对任意可测体$K$，其等周系数$\psi_K$的量级为$O(n^{-1/2})$。作为推论，我们得出轴对齐立方体本质上是“最大化”$\ell_0$等周系数的：存在正常数$q>0$使得对任意轴对齐立方体$\mathcal{C}$和任意可测集$K$，均有$\psi_K \leq q \cdot \psi_{\mathcal{C}}$。最后，我们将所得结果直接应用于凸体均匀采样中坐标游走算法的混合时间估计。