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Persistent Topology studies topological features of shapes by analyzing thelower level sets of suitable functions, called filtering functions, andencoding the arising information in a parameterized version of the Bettinumbers, i.e. the ranks of persistent homology groups. Initially introduced byconsidering real-valued filtering functions, Persistent Topology has beensubsequently generalized to a multidimensional setting, i.e. to the case of$\R^n$-valued filtering functions, leading to studying the ranks ofmultidimensional homology groups. In particular, a multidimensional matchingdistance has been defined, in order to compare these ranks. The definition ofthe multidimensional matching distance is based on foliating the domain of theranks of multidimensional homology groups by a collection of half-planes, andhence it formally depends on a subset of $\R^n\times\R^n$ inducing aparameterization of these half-planes. It happens that it is possible to choosethis subset in an infinite number of different ways. In this paper we show thatthe multidimensional matching distance is actually invariant with respect tosuch a choice.

Invariance properties of the multidimensional matching distance in Persistent Topology and Homology