  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.
