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

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.