  It is a well known fact that in $\mathbb{R}^n$ a subset of minimal perimeter
$L$ among all sets of a given volume is also a set of maximal volume among all
sets of the same perimeter $L$. This is called the reciprocity principle for
isoperimetric problems. The aim of this note is to prove this relation in the
case where the class of admissible sets is restricted to the subsets of some
subregion $G\subsetneq\mathbb{R}^n$. Furthermore, we give a characterization of
those (unbounded) convex subsets of $\mathbb{R}^2$ in which the isoperimetric
problem has a solution. The perimeter that we consider is the one relative to
$\mathbb{R}^n$.
