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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>. |
| | 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> |
| | .<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]] |
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]
See also
References
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- ↑ 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.
- ↑ 2.0 2.1 Pietro Donatini, Patrizio Frosini, Natural pseudodistances between closed manifolds, Forum Mathematicum, 16(5):695-715, 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.