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\$ .^[1]

Main properties

It can be proved ^[2] 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\$.^[3] If $M\$ and $N\$ are curves, the number $k\$ can be assumed to be $1\$ or $2\$.^[4] 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\$.^[2]

See also

• Fréchet distance
• Size function
• Size functor
• Size homotopy group

References

1. Patrizio Frosini, Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society, 6:455-464, 1999.
2. Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed manifolds, Forum Mathematicum, 16(5):695-715, 2004.
3. Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed surfaces, Journal of the European Mathematical Society, 9(2):231–253, 2007.
4. Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed curves, Forum Mathematicum, 21(6):981–999, 2009.