On the ℓ0 Isoperimetric Coefficient of Measurable Sets

In this paper we prove that the ℓ0 isoperimetric coefficient for any axis-aligned cubes, psiC, is Theta(n-1/2) and that the isoperimetric coefficient for any measurable body K, psiK, is of order O(n-1/2). As a corollary we deduce that axis-aligned cubes essentially "maximize" the ℓ0 isoperimetric coefficient: There exists a positive constant q > 0 such that psiK ≤ q · psiC, whenever 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.