 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 ^{[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 . If and are surfaces, the number can be assumed to be , or ^{[3]}. If and are curves, the number can be assumed to be or ^{[4]}. If an optimal homeomorphism exists (i.e., ), then can be assumed to be ^{[2]}.
References
 ^ Patrizio Frosini, Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society  Simon Stevin, 6:455464, 1999.
 ^ ^{a} ^{b} Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed manifolds, Forum Mathematicum, 16(5):695715, 2004.
 ^ Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed surfaces, Journal of the European Mathematical Society, 9(2):231–253, 2007.
 ^ Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed curves, Forum Mathematicum, 21(6):981–999, 2009.
See also
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