Core Concepts
The ℓ0 isoperimetric coefficient of axis-aligned cubes is Θ(n^(-1/2), and the ℓ0 isoperimetric coefficient of any measurable set is O(n^(-1/2)).
Abstract
The paper proves the following key results:
The ℓ0 isoperimetric coefficient of axis-aligned cubes, denoted as ψ_C, is Θ(n^(-1/2)). This is shown by first proving a combinatorial analogue on the discrete hypercube and then extending it to the continuous cube.
The ℓ0 isoperimetric coefficient of any measurable set K, denoted as ψ_K, is O(n^(-1/2)). This is shown by extending the approach used for the cube to general measurable sets. The key idea is to partition the set K into axis-disjoint subsets using a random splitting plane, and then bound the ℓ0 boundary of the resulting subsets.
As a corollary, the author shows that axis-aligned cubes essentially "maximize" the ℓ0 isoperimetric coefficient: There exists a positive constant q > 0 such that ψ_K ≤ q * ψ_C for any axis-aligned cube C and any measurable set K.
The improved bounds on the ℓ0 isoperimetry immediately imply improved mixing time bounds for the Coordinate-Hit-and-Run Markov chain used for sampling from convex bodies.