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 pe
rsistent 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 $R^ntimesR^n$ 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.
Some new results about multidimensional Topological Persistence are presented, proving that the discontinuity points of a k-dimensional size function are necessarily related to the pseudocritical or special values of the associated measuring function.
The concept of natural pseudo-distance has proven to be a powerful tool for measuring the dissimilarity between topological spaces endowed with continuous real-valued functions. Roughly speaking, the natural pseudo-distance is defined as the infimum
of the change of the functions values, when moving from one space to the other through homeomorphisms, if possible. In this paper, we prove the first available result about the existence of optimal homeomorphisms between closed curves, i.e. inducing a change of the function that equals the natural pseudo-distance.
