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| In [[mathematics]], the '''Menger curvature''' of a triple of points in ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> is the [[Multiplicative inverse|reciprocal]] of the [[radius]] of the circle that passes through the three points. It is named after the [[Austria]]n-[[United States|American]] [[mathematician]] [[Karl Menger]].
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| ==Definition==
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| Let ''x'', ''y'' and ''z'' be three points in '''R'''<sup>''n''</sup>; for simplicity, assume for the moment that all three points are distinct and do not lie on a single straight line. Let Π ⊆ '''R'''<sup>''n''</sup> be the [[Plane (mathematics)|Euclidean plane]] spanned by ''x'', ''y'' and ''z'' and let ''C'' ⊆ Π be the unique [[circle|Euclidean circle]] in Π that passes through ''x'', ''y'' and ''z'' (the [[Circumscribed circle|circumcircle]] of ''x'', ''y'' and ''z''). Let ''R'' be the radius of ''C''. Then the '''Menger curvature''' ''c''(''x'', ''y'', ''z'') of ''x'', ''y'' and ''z'' is defined by
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| :<math>c (x, y, z) = \frac1{R}.</math>
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| If the three points are [[Line (geometry)|collinear]], ''R'' can be informally considered to be +∞, and it makes rigorous sense to define ''c''(''x'', ''y'', ''z'') = 0. If any of the points ''x'', ''y'' and ''z'' are coincident, again define ''c''(''x'', ''y'', ''z'') = 0.
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| Using the well-known formula relating the side lengths of a [[triangle]] to its area, it follows that
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| :<math>c (x, y, z) = \frac1{R} = \frac{4 A}{| x - y | | y - z | | z - x |},</math>
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| where ''A'' denotes the area of the triangle spanned by ''x'', ''y'' and ''z''.
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| Another way of computing Menger curvature is the identity
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| :<math> c(x,y,z)=\frac{2\sin \angle xyz}{|x-z|}</math>
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| where <math>\angle xyz</math> is the angle made at the ''y''-corner of the triangle spanned by ''x'',''y'',''z''.
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| Menger curvature may also be defined on a general [[metric space]]. If ''X'' is a metric space and ''x'',''y'', and ''z'' are distinct points, let ''f'' be an [[isometry]] from <math>\{x,y,z\}</math> into <math>\mathbb{R}^{2}</math>. Define the Menger curvature of these points to be
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| :<math> c_{X} (x,y,z)=c(f(x),f(y),f(z)).</math>
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| Note that ''f'' need not be defined on all of ''X'', just on ''{x,y,z}'', and the value ''c''<sub>''X''</sub> ''(x,y,z)'' is independent of the choice of ''f''.
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| ==Integral Curvature Rectifiability==
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| Menger curvature can be used to give quantitative conditions for when sets in <math> \mathbb{R}^{n} </math> may be [[Rectifiable set|rectifiable]]. For a [[Borel measure]] <math>\mu</math> on a Euclidean space <math> \mathbb{R}^{n}</math> define
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| : <math> c^{p}(\mu)=\int\int\int c(x,y,z)^{p}d\mu(x)d\mu(y)d\mu(z).</math>
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| * A Borel set <math> E\subseteq \mathbb{R}^{n} </math> is rectifiable if <math> c^{2}(H^{1}|_{E})<\infty</math>, where <math> H^{1}|_{E} </math> denotes one-dimensional [[Hausdorff measure]] restricted to the set <math> E</math>.<ref>{{cite journal
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| | last = Leger | first = J. | title =Menger curvature and rectifiability
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| | journal = Annals of Mathematics | volume = 149 | pages = 831–869 | year = 1999
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| | url = http://www.emis.de/journals/Annals/149_3/leger.pdf
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| | issue = 3
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| | doi = 10.2307/121074
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| | jstor = 121074
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| | publisher = Annals of Mathematics
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| }}</ref>
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| The basic intuition behind the result is that Menger curvature measures how straight a given triple of points are (the smaller <math> c(x,y,z)\max\{|x-y|,|y-z|,|z-y|\}</math> is, the closer x,y, and z are to being collinear), and this integral quantity being finite is saying that the set E is flat on most small scales. In particular, if the power in the integral is larger, our set is smoother than just being rectifiable<ref>{{cite journal
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| | last = Pawl Strzelecki, Marta Szumanska, Heiko von der Mosel | title =Regularizing and self-avoidance effects of integral Menger curvature
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| | journal = Institut f¨ur Mathematik | eprint = 29 year = 2008
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| }}</ref>
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| * Let <math> p>3</math>, <math> f:S^{1}\rightarrow \mathbb{R}^{n}</math> be a homeomorphism and <math>\Gamma=f(S^{1})</math>. Then <math> f\in C^{1,1-\frac{3}{p}}(S^{1})</math> if <math> c^{p}(H^{1}|_{\Gamma})<\infty</math>.
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| * If <math> 0<H^{s}(E)<\infty</math> where <math> 0<s\leq\frac{1}{2}</math>, and <math> c^{2s}(H^{s}|_{E})<\infty</math>, then <math> E</math> is rectifiable in the sense that there are countably many <math>C^{1}</math> curves <math>\Gamma_{i}</math> such that <math> H^{s}(E\backslash \bigcup\Gamma_{i})=0</math>. The result is not true for <math> \frac{1}{2}<s<1</math>, and <math> c^{2s}(H^{s}|_{E})=\infty</math> for <math> 1<s\leq n</math>.:<ref>{{cite journal
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| | last = Yong Lin and Pertti Mattila | title =Menger curvature and <math> C^{1}</math> regularity of fractals
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| | journal = Proceedings of the American Mathematical Society | volume = 129 | pages = 1755–1762 | year = 2000
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| | url = http://www.ams.org/proc/2001-129-06/S0002-9939-00-05814-7/S0002-9939-00-05814-7.pdf
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| | issue = 6
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| }}</ref>
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| In the opposite direction, there is a result of Peter Jones:<ref>{{cite book
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| | last = Pajot | first = H. | title =Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral
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| | publisher = Springer| year = 2000
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| | isbn = 3-540-00001-1
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| }}</ref>
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| * If <math>E\subseteq\Gamma\subseteq\mathbb{R}^{2}</math>, <math> H^{1}(E)>0</math>, and <math>\Gamma</math> is rectifiable. Then there is a positive Radon measure <math>\mu</math> supported on <math>E</math> satisfying <math> \mu B(x,r)\leq r</math> for all <math>x\in E</math> and <math>r>0</math> such that <math>c^{2}(\mu)<\infty</math> (in particular, this measure is the [[Frostman's lemma|Frostman measure]] associated to E). Moreover, if <math>H^{1}(B(x,r)\cap\Gamma)\leq Cr</math> for some constant ''C'' and all <math> x\in \Gamma</math> and ''r>0'', then <math> c^{2}(H^{1}|_{E})<\infty</math>. This last result follows from the [[Analyst's Traveling Salesman Theorem]].
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| Analogous results hold in general metric spaces:<ref>{{cite journal
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| | last = Schul | first = Raanan | title =Ahlfors-regular curves in metric spaces
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| | journal = Annales Academiæ Scientiarum Fennicæ | volume = 32 | pages = 437–460 | year = 2007
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| | url =http://www.acadsci.fi/mathematica/Vol32/vol32pp437-460.pdf
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| }}</ref>
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| ==See also==
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| * [[Curvature of a measure|Menger-Melnikov curvature of a measure]]
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| ==External links==
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| * {{cite web
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| | url = http://www.lems.brown.edu/vision/people/leymarie/Notes/CurvSurf/MengerCurv/index.html
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| | title = Notes on Menger Curvature
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| | last = Leymarie
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| | first = F.
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| | accessdate = 2007-11-19
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| |date=September 2003
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| |archiveurl = http://web.archive.org/web/20070821103738/http://www.lems.brown.edu/vision/people/leymarie/Notes/CurvSurf/MengerCurv/index.html <!-- Bot retrieved archive --> |archivedate = 2007-08-21}}
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| ==References==
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| <references/>
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| * {{cite journal
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| | last = Tolsa
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| | first = Xavier
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| | title = Principal values for the Cauchy integral and rectifiability
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| | journal = [[Proceedings of the American Mathematical Society|Proc. Amer. Math. Soc.]]
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| | volume = 128
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| | year = 2000
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| | pages = 2111–2119
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| | doi = 10.1090/S0002-9939-00-05264-3
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| | issue = 7
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| }}
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| [[Category:Curvature (mathematics)]]
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| [[Category:Multi-dimensional geometry]]
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The name of the writer is Jayson. One of the extremely best things in the globe for him is performing ballet and he'll be starting some thing else alongside with it. North Carolina is the location he loves most but now he is considering other options. My day job is a travel agent.
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