The Edge-Isoperimetric Problem in $(\mathbb{N}^2,\infty)$

TL;DR

This paper solves the discrete edge-isoperimetric problem on the infinite grid with the ll_{ot} metric, characterizing optimal sets, their perimeter growth, and the behavior of minimal perimeter values across volumes.

Contribution

It introduces a geometric form for optimal sets, establishes asymptotic perimeter growth, and provides bounds, effectively solving the isoperimetric inequality on this graph.

Findings

Optimal sets are of a specific geometric form.

Perimeter grows asymptotically as 2rac{1}{2}  rac{1}{2} rac{7x}{2}.

Existence of arbitrarily long intervals with constant minimal perimeter.

Abstract

We consider the edge-isoperimetric problem on the graph of the infinite grid $\mathbb{N}^{2}$ in the $\ell_{\infty}$ metric. We first show that the solutions are not nested, so that techniques other than compressions have to be used. We then show that for any given volume of sets in $\mathbb{N}^{2}$, there exists an optimal set of a specific geometric form and describe this form. We continue on to prove that the optimal perimeter has asymptotic growth rate $2\sqrt{7x}$ as a function of the volume and obtain upper and lower bounds for the optimal perimeter which are within the small additive constant of $\frac{35}{2}$ of one another, thus effectively solving the discrete isoperimetric inequality on this graph. Finally, we prove that there exist arbitrarily long consecutive values of the volume for which the minimum perimeter is the same.