In size theory, the natural pseudodistance between two size pairs
‖ φ − ψ ∘ h
varies in the set of all homeomorphisms from the manifold
is the supremum norm.
are not homeomorphic, then the natural pseudodistance is defined to be
closed manifolds and the measuring functions
φ , ψ
Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from
The concept of natural pseudodistance can be easily extended to size pairs where the measuring function
takes values in
of all homeomorphisms of
can be replaced in the definition of natural pseudodistance by a subgroup
, so obtaining the concept of natural pseudodistance with respect to the group
[2][3] Lower bounds and approximations of the natural pseudodistance with respect to the group
can be obtained both by means of
-invariant persistent homology[4] and by combining classical persistent homology with the use of G-equivariant non-expansive operators.
[2][3] It can be proved [5] that the natural pseudodistance always equals the Euclidean distance between two critical values of the measuring functions (possibly, of the same measuring function) divided by a suitable positive integer
are surfaces, the number
can be assumed to be
are curves, the number
can be assumed to be
[7] If an optimal homeomorphism
‖ φ − ψ ∘
inf
‖ φ − ψ ∘ h
can be assumed to be
[5] The research concerning optimal homeomorphisms is still at its very beginning .
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