Menger curvature

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

$c (x, y, z) = \frac1{R}.$

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

$c (x, y, z) = \frac1{R} = \frac{4 A}{| x - y | | y - z | | z - x |},$

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

Another way of computing Menger curvature is the identity

$c(x,y,z)=\frac{2\sin \angle xyz}{|x-z|}$

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 $\mathbb{R}^{2}$. Define the Menger curvature of these points to be

c_X(x,y,z) = c(f(x),f(y),f(z)).

Note that f need not be defined on all of X, just on {x,y,z}, 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 $\mathbb{R}^{n}$ may be rectifiable. For a Borel measure μ on a Euclidean space $\mathbb{R}^{n}$ define

$c^{p}(\mu)=\int\int\int c(x,y,z)^{p}d\mu(x)d\mu(y)d\mu(z).$

• A Borel set $E\subseteq \mathbb{R}^{n}$ is rectifiable if $c^{2}(H^{1}|_{E})<\infty$, where H¹ | _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, $f:S^{1}\rightarrow \mathbb{R}^{n}$ be a homeomorphism and Γ = f(S¹). Then $f\in C^{1,1-\frac{3}{p}}(S^{1})$ if $c^{p}(H^{1}|_{\Gamma})<\infty$.

• If $0<H^{s}(E)<\infty$ where $0<s\leq\frac{1}{2}$, and $c^{2s}(H^{s}|_{E})<\infty$, then E is rectifiable in the sense that there are countably many C¹ curves Γ_i such that $H^{s}(E\backslash \bigcup\Gamma_{i})=0$. The result is not true for $\frac{1}{2}<s<1$, and $c^{2s}(H^{s}|_{E})=\infty$ for $1<s\leq n$.^[3]:

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

• If $E\subseteq\Gamma\subseteq\mathbb{R}^{2}$, H¹(E) > 0, and Γ is rectifiable. Then there is a positive Radon measure μ supported on E satisfying $\mu B(x,r)\leq r$ for all $x\in E$ and r > 0 such that $c^{2}(\mu)<\infty$ (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 $c^{2}(H^{1}|_{E})<\infty$. 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

• Leymarie, F. (September 2003). "Notes on Menger Curvature". Archived from the original on 2007-08-21. http://web.archive.org/web/20070821103738/http://www.lems.brown.edu/vision/people/leymarie/Notes/CurvSurf/MengerCurv/index.html. Retrieved 2007-11-19. 

References

1. ^ Leger, J. (1999). "Menger curvature and rectifiability". Annals of Mathematics (Annals of Mathematics) 149 (3): 831–869. doi:10.2307/121074. JSTOR 121074. http://www.emis.de/journals/Annals/149_3/leger.pdf. 
2. ^ Pawl Strzelecki, Marta Szumanska, Heiko von der Mosel. "Regularizing and self-avoidance effects of integral Menger curvature". Institut f¨ur Mathematik. 
3. ^ Yong Lin and Pertti Mattila (2000). "Menger curvature and C¹ regularity of fractals". Proceedings of the American Mathematical Society 129 (6): 1755–1762. http://www.ams.org/proc/2001-129-06/S0002-9939-00-05814-7/S0002-9939-00-05814-7.pdf. 
4. ^ Pajot, H. (2000). Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral. Springer. ISBN 3540000011. 
5. ^ Schul, Raanan (2007). "Ahlfors-regular curves in metric spaces". Annales Academiæ Scientiarum Fennicæ 32: 437–460. http://www.acadsci.fi/mathematica/Vol32/vol32pp437-460.pdf. 

• Tolsa, Xavier (2000). "Principal values for the Cauchy integral and rectifiability". Proc. Amer. Math. Soc. 128 (7): 2111–2119. doi:10.1090/S0002-9939-00-05264-3.