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


The concept of natural pseudodistance can be easily extended to [[size pair]]s where the measuring function <math>\varphi\ </math> takes values in <math>\mathbb{R}^m\ </math>
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.<ref name="FroMu99">Patrizio Frosini, Michele Mulazzani, ''Size homotopy groups for computation of natural size distances'', Bulletin of the Belgian Mathematical Society - Simon Stevin, 6:455-464, 1999.</ref>
 
==Main properties==
It can be proved <ref name="DoFro04">Pietro Donatini, Patrizio Frosini, ''Natural pseudodistances between closed manifolds'', Forum Mathematicum, 16(5):695-715, 2004.</ref>
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 <math>k\ </math>.
If <math>M\ </math> and <math>N\ </math> are surfaces, the number <math>k\ </math> can be assumed to be <math>1\ </math>, <math>2\ </math> or <math>3\ </math>.<ref>Pietro Donatini, Patrizio Frosini, ''Natural pseudodistances between closed surfaces'',
Journal of the European Mathematical Society, 9(2):231–253, 2007.</ref> If <math>M\ </math> and <math>N\ </math> are curves, the number <math>k\ </math> can be assumed to be <math>1\ </math> or <math>2\ </math>.<ref>Pietro Donatini, Patrizio Frosini, ''Natural pseudodistances between closed curves'', Forum Mathematicum, 21(6):981–999, 2009.</ref>
If an optimal homeomorphism <math>\bar h\ </math> exists (i.e., <math>\|\varphi-\psi\circ \bar h\|_\infty=\inf_h \|\varphi-\psi\circ h\|_\infty\ </math>), then <math>k\ </math> can be assumed to be <math>1\ </math>.<ref name="DoFro04"/>
 
==See also==
* [[Fréchet surface|Fréchet distance]]
* [[Size function]]
* [[Size functor]]
* [[Size homotopy group]]
 
==References==
{{reflist}}
 
[[Category:Differential geometry]]

Latest revision as of 03:04, 8 January 2015

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