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| In the [[mathematics|mathematical]] study of [[metric spaces]], one can consider the [[arclength]] of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from one point to another along a path whose arclength is equal to (or very close to) that distance. The distance between two points of a metric space relative to the '''intrinsic metric''' is defined as the [[infimum]] of the length of all paths from one point to the other. A metric space is a '''length metric space''' if the intrinsic metric agrees with the original metric of the space.
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| ==Definitions==
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| Let <math>(M, d)\,</math> be a [[metric space]]. We define a new metric <math>d_I\,</math> on <math>M\,</math>, known as the '''induced intrinsic metric''', as follows:
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| <math>d_I(x,y)\,</math> is the [[infimum]] of the lengths of all paths from <math>x\,</math> to <math>y\,</math>.
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| Here, a ''path'' from <math>x\,</math> to <math>y\,</math> is a [[continuous map]]
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| :<math>\gamma : [0,1] \rightarrow M</math> | |
| with <math>\gamma(0) = x\,</math> and <math>\gamma(1) = y\,</math>. The ''length'' of such a path is defined as explained for [[rectifiable curve]]s. We set <math>d_I(x,y) =\infty</math> if there is no path of finite length from <math>x\,</math> to <math>y\,</math>. If
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| :<math>d_I(x,y)=d(x,y)\, </math>
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| for all points <math>x\,</math> and <math>y\,</math> in <math>M\,</math>, we say that <math>(M, d)\,</math> is a '''length space''' or a '''path metric space''' and the metric <math>d\,</math> is '''intrinsic'''.
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| We say that the metric <math>d\,</math> has '''approximate midpoints''' if for any <math>\varepsilon>0</math> and any pair of points <math>x\,</math> and <math>y\,</math> in <math>M\,</math> there exists <math>c\,</math> in <math>M\,</math> such that <math>d(x,c)\,</math> and <math>d(c,y)\,</math> are both smaller than
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| :<math>{d(x,y)}/{2} + \varepsilon</math>.
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| ==Examples==
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| * [[Euclidean space]] '''R'''<sup>''n''</sup> with the ordinary Euclidean metric is a path metric space. '''R'''<sup>''n''</sup> - {0} is as well.
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| * The [[unit circle]] S<sup>1</sup> with the metric inherited from the Euclidean metric of '''R'''<sup>2</sup> (the '''chordal metric''') is not a path metric space. The induced intrinsic metric on S<sup>1</sup> measures distances as [[angle]]s in [[radian]]s, and the resulting length metric space is called the [[Riemannian circle]]. In two dimensions, the chordal metric on the [[sphere]] is not intrinsic, and the induced intrinsic metric is given by the [[great-circle distance]].
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| * Every [[Riemannian manifold]] can be turned into a path metric space by defining the distance of two points as the infimum of the lengths of continuously differentiable curves connecting the two points. (The Riemannian structure allows one to define the length of such curves.) Analogously, other manifolds in which a length is defined included [[Finsler manifold]]s and [[sub-Riemannian manifold]]s.
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| * Any [[complete metric space|complete]] and [[convex metric space]] is a length metric space {{harv|Khamsi|Kirk|2001|loc=Theorem 2.16}}, a result of [[Karl Menger]]. The converse does not hold in general, however: there are length metric spaces which are not convex.
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| == Properties ==
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| *In general, we have ''d'' ≤ ''d<sub>l</sub>'' and the [[topological space|topology]] defined by ''d<sub>l</sub>'' is therefore always [[finer topology|finer]] than or equal to the one defined by ''d''.
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| *The space (''M'', ''d''<sub>''l''</sub>) is always a path metric space (with the caveat, as mentioned above, that ''d''<sub>''l''</sub> can be infinite).
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| *The metric of a length space has approximate midpoints. Conversely, every [[complete space|complete]] metric space with approximate midpoints is a length space.
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| *The [[Hopf–Rinow theorem]] states that if a length space <math>(M,d)</math> is [[complete space|complete]] and [[locally compact]] then any two points in <math>M</math> can be connected by a [[geodesic|minimizing geodesic]] and all bounded [[closed set]]s in <math>M</math> are [[compact set|compact]].
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| ==References==
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| * {{citation|authorlink=Mikhail Gromov (mathematician)|first=Mikhail|last=Gromov|title=[[Metric Structures for Riemannian and Non-Riemannian Spaces]]|series=Progress in Math.|volume=152|publisher=Birkhäuser|year=1999|isbn=0-8176-3898-9}}
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| *{{citation
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| | authorlink1 = Mohamed Amine Khamsi
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| | last1 = Khamsi
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| | first1 = Mohamed A.
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| | authorlink2 = William Arthur Kirk
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| | last2 = Kirk
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| | first2 = William A.
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| | title = An Introduction to Metric Spaces and Fixed Point Theory
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| | publisher = Wiley-IEEE
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| | date = 2001
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| | pages =
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| | isbn = 0-471-41825-0
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| }}
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| [[Category:Metric geometry]]
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