Natural pseudodistance


In size theory, the natural pseudodistance between two size pairs, is the value, where varies in the set of all homeomorphisms from the manifold to the manifold and is the supremum norm. If and are not homeomorphic, then the natural pseudodistance is defined to be.
It is usually assumed that, are closed manifolds and the measuring functions are. Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from to.
The concept of natural pseudodistance can be easily extended to size pairs where the measuring function takes values in
. When, the group of all homeomorphisms of can be replaced in the definition of natural pseudodistance by a subgroup of, so obtaining the concept of natural pseudodistance with respect to the group
. Lower bounds and approximations of the natural pseudodistance with respect to the group can be obtained both by means of -invariant persistent homology and by combining classical persistent homology with the use of G-equivariant non-expansive operators

Main properties

It can be proved
that the natural pseudodistance always equals the Euclidean distance between two critical values of the measuring functions divided by a suitable positive integer.
If and are surfaces, the number can be assumed to be, or. If and are curves, the number can be assumed to be or.
If an optimal homeomorphism exists, then can be assumed to be. The research concerning optimal homeomorphisms is still at its very beginning