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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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