  Persistent Topology studies topological features of shapes by analyzing the
lower level sets of suitable functions, called filtering functions, and
encoding the arising information in a parameterized version of the Betti
numbers, i.e. the ranks of persistent homology groups. Initially introduced by
considering real-valued filtering functions, Persistent Topology has been
subsequently generalized to a multidimensional setting, i.e. to the case of
$\R^n$-valued filtering functions, leading to studying the ranks of
multidimensional homology groups. In particular, a multidimensional matching
distance has been defined, in order to compare these ranks. The definition of
the multidimensional matching distance is based on foliating the domain of the
ranks of multidimensional homology groups by a collection of half-planes, and
hence it formally depends on a subset of $\R^n\times\R^n$ inducing a
parameterization of these half-planes. It happens that it is possible to choose
this subset in an infinite number of different ways. In this paper we show that
the multidimensional matching distance is actually invariant with respect to
such a choice.
