Natural pseudodistance

In size theory, the natural pseudodistance between two size pairs $$(M,\varphi:M\to \mathbb{R})\ $$, $$(N,\psi:N\to \mathbb{R})\ $$ is the value $$\inf_h \|\varphi-\psi\circ h\|_\infty\ $$, where $$h\ $$ varies in the set of all homeomorphisms from the manifold $$M\ $$ to the manifold $$N\ $$ and $$\|\cdot\|_\infty\ $$ is the supremum norm. If $$M\ $$ and $$N\ $$ are not homeomorphic, then the natural pseudodistance is defined to be $$\infty\ $$. It is usually assumed that $$M\ $$, $$N\ $$ are $$C^1\ $$ closed manifolds and the measuring functions $$\varphi,\psi\ $$ are $$C^1\ $$. Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from $$M\ $$ to $$N\ $$.

The concept of natural pseudodistance can be easily extended to size pairs where the measuring function $$\varphi\ $$ takes values in $$\mathbb{R}^m\ $$ . When $$M=N\ $$, the group $$H\ $$ of all homeomorphisms of $$M\ $$ can be replaced in the definition of natural pseudodistance by a subgroup $$G\ $$ of $$H\ $$, so obtaining the concept of natural pseudodistance with respect to the group $$G\ $$. Lower bounds and approximations of the natural pseudodistance with respect to the group $$G\ $$ can be obtained both by means of $$G$$-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 (possibly, of the same measuring function) divided by a suitable positive integer $$k\ $$. If $$M\ $$ and $$N\ $$ are surfaces, the number $$k\ $$ can be assumed to be $$1\ $$, $$2\ $$ or $$3\ $$. If $$M\ $$ and $$N\ $$ are curves, the number $$k\ $$ can be assumed to be $$1\ $$ or $$2\ $$. If an optimal homeomorphism $$\bar h\ $$ exists (i.e., $$\|\varphi-\psi\circ \bar h\|_\infty=\inf_h \|\varphi-\psi\circ h\|_\infty\ $$), then $$k\ $$ can be assumed to be $$1\ $$. The research concerning optimal homeomorphisms is still at its very beginning .