Monotonicity of the optimal perimeter in isoperimetric problems on ${\mathbb{Z}}^{k} \times {\mathbb{N}}^{d}$

TL;DR

This paper establishes that, under certain conditions, the perimeter of optimal sets in isoperimetric problems on specific lattice structures increases monotonically with volume, providing new insights into their geometric properties.

Contribution

It proves the monotonicity of the perimeter in isoperimetric problems on lattices of the form Z^k x N^d, even when solutions are not nested, expanding understanding of optimal set properties.

Findings

Perimeter is monotonically increasing with volume under natural assumptions.

Monotonicity applies to vertex- and edge-isoperimetric problems in ℓ_p distances.

Existence of arbitrarily long volume intervals with constant minimal boundary in N^2 under ℓ_∞ distance.

Abstract

We prove general theorems for isoperimetric problems on lattices of the form ${\mathbb{Z}}^{k} \times {\mathbb{N}}^{d}$ which state that the perimeter of the optimal set is a monotonically increasing function of the volume under certain natural assumptions, such as local symmetry or being induced by an $\ell_p$-norm. The proved monotonicity property is surprising considering that solutions are not always nested (and consequently standard techniques such as compressions do not apply). The monotonicity results of this note apply in particular to vertex- and edge-isoperimetric problems in the $\ell_p$ distances and can be used as a tool to elucidate properties of optimal sets. As an application, we consider the edge-isoperimetric inequality on the graph ${\N}^2$ in the $\ell_\infty$-distance. We show that there exist arbitrarily long consecutive values of the volume for which the minimum boundary is the same.