Menger curvature explained

In mathematics, the Menger curvature of a triple of points in n-dimensional Euclidean space Rⁿ is the reciprocal of the radius of the circle that passes through the three points. It is named after the Austrian-American mathematician Karl Menger.

Definition

Let x, y and z be three points in Rⁿ; for simplicity, assume for the moment that all three points are distinct and do not lie on a single straight line. Let Π ⊆ Rⁿ be the Euclidean plane spanned by x, y and z and let C ⊆ Π be the unique Euclidean circle in Π that passes through x, y and z (the 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

c (x, y, z) = (1 {R}.)/()

If the three points are 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.

Using the well-known formula relating the side lengths of a triangle to its area, it follows that

c (x, y, z) = (1 {R})/(=) (4 A)/(|x - y ||y - z ||z - x |) ,

where A denotes the area of the triangle spanned by x, y and z.

Another way of computing Menger curvature is the identity

c(x,y,z) = (2 \sin \angle xyz)/(|x - z|) where \angle xyz is the angle made at the y-corner of the triangle spanned by x,y,z.

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

\{x,y,z\} into R ^{2} . Define the Menger curvature of these points to bec _{X} (x,y,z) = c(f(x),f(y),f(z)).

Note that f need not be defined on all of X, just on , and the value c_X (x,y,z) is independent of the choice of f.

Integral Curvature Rectifiability

Menger curvature can be used to give quantitative conditions for when sets in

R ^{n} may be rectifiable. For a Borel measure \mu on a Euclidean space R ^{n} definec ^{p} (\mu) = \int\int\int c(x,y,z) ^{p} d\mu(x)d\mu(y)d\mu(z).

• A Borel set

E\subseteq R ^{n} is rectifiable if c ^{2} (H ^{1} | _{E} )< inf ty , where H ^{1} | _{E} denotes one-dimensional Hausdorff measure restricted to the set E .^[1]

The basic intuition behind the result is that Menger curvature measures how straight a given triple of points are (the smaller

c(x,y,z) max \{|x - y|,|y - z|,|z - y|\} 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^[2]

• Let

p> 3 , f:S ^{1} → R ^{n} be a homeomorphism and \Gamma = f(S ^{1} ) . Then f\in C ^{1 , 1- (3)/(p)} (S ^{1} ) if c ^{p} (H ^{1} | _{\Gamma} )< inf ty .

• If

0 <H ^{s} (E)< inf ty where 0 <s\leq (1)/(2) , and c ^{2s} (H ^{s} | _{E} )< inf ty , then E is rectifiable in the sense that there are countably many C ^{1} curves \Gamma _{i} such that H ^{s} (E\backslash c up\Gamma _{i} ) =0 . The result is not true for (1)/(2) <s< 1 , and c ^{2s} (H ^{s} | _{E} ) =inf ty for 1 <s\leq n .:^[3]

In the opposite direction, there is a result of Peter Jones:^[4]

• If

E\subseteq\Gamma\subseteq R ^{2} , H ^{1} (E)> 0 , and \Gamma is rectifiable. Then there is a positive Radon measure \mu supported on E satisfying \mu B(x,r)\leq r for all x\in E and r> 0 such that c ^{2} (\mu)< inf ty (in particular, this measure is the Frostman measure associated to E). Moreover, if H ^{1} (B(x,r)\cap\Gamma)\leq Cr for some constant C and all x\in \Gamma and r>0, then c ^{2} (H ^{1} | _{E} )< inf ty . This last result follows from the Analyst's Traveling Salesman Theorem.

Analogous results hold in general metric spaces:^[5]

See also

• Menger-Melnikov curvature of a measure

External links

• Web site: Leymarie . F. . September 2003 . Notes on Menger Curvature . 2007-11-19 . https://web.archive.org/web/20070821103738/http://www.lems.brown.edu/vision/people/leymarie/Notes/CurvSurf/MengerCurv/index.html . 2007-08-21 . dead.

References

1. Leger . J. . 1999 . Menger curvature and rectifiability . . 149 . 3 . 831–869 . math/9905212 . 10.2307/121074 . 121074 . 216176.
2. Strzelecki . Paweł . Szumańska . Marta . von der Mosel . Heiko . 2010 . Regularizing and self-avoidance effects of integral Menger curvature . . 9 . 1 . 145–187.
3. Lin . Yong . Mattila . Pertti . 2000 . Menger curvature and C¹ regularity of fractals . . 129 . 6 . 1755–1762 . 10.1090/s0002-9939-00-05814-7 . free.
4. Book: Pajot, H. . 2000 . Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral . Springer . 3-540-00001-1.
5. Schul . Raanan . 2007 . Ahlfors-regular curves in metric spaces . . 32 . 437–460.

• Tolsa. Xavier . 2000 . Principal values for the Cauchy integral and rectifiability . . 128 . 7 . 2111–2119 . 10.1090/S0002-9939-00-05264-3 . free.