On the $\ell_0$ Isoperimetric Coefficient of Measurable Sets

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.