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-United States 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

[math]\displaystyle{ c (x, y, z) = \frac1{R}. }[/math]

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

[math]\displaystyle{ c (x, y, z) = \frac1{R} = \frac{4 A}{| x - y | | y - z | | z - x |}, }[/math]

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

Another way of computing Menger curvature is the identity

[math]\displaystyle{ c(x,y,z)=\frac{2\sin \angle xyz}{|x-z|} }[/math]

where [math]\displaystyle{ \angle xyz }[/math] 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 [math]\displaystyle{ \{x,y,z\} }[/math] into [math]\displaystyle{ \mathbb{R}^{2} }[/math]. Define the Menger curvature of these points to be

[math]\displaystyle{ c_{X} (x,y,z)=c(f(x),f(y),f(z)). }[/math]

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 [math]\displaystyle{ \mathbb{R}^{n} }[/math] may be rectifiable. For a Borel measure [math]\displaystyle{ \mu }[/math] on a Euclidean space [math]\displaystyle{ \mathbb{R}^{n} }[/math] define

[math]\displaystyle{ c^{p}(\mu)=\int\int\int c(x,y,z)^{p}d\mu(x)d\mu(y)d\mu(z). }[/math]

• A Borel set [math]\displaystyle{ E\subseteq \mathbb{R}^{n} }[/math] is rectifiable if [math]\displaystyle{ c^{2}(H^{1}|_{E})\lt \infty }[/math], where [math]\displaystyle{ H^{1}|_{E} }[/math] denotes one-dimensional Hausdorff measure restricted to the set [math]\displaystyle{ E }[/math].

The basic intuition behind the result is that Menger curvature measures how straight a given triple of points are (the smaller [math]\displaystyle{ 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

• Let [math]\displaystyle{ p\gt 3 }[/math], [math]\displaystyle{ f:S^{1}\rightarrow \mathbb{R}^{n} }[/math] be a homeomorphism and [math]\displaystyle{ \Gamma=f(S^{1}) }[/math]. Then [math]\displaystyle{ f\in C^{1,1-\frac{3}{p}}(S^{1}) }[/math] if [math]\displaystyle{ c^{p}(H^{1}|_{\Gamma})\lt \infty }[/math].
• If [math]\displaystyle{ 0\lt H^{s}(E)\lt \infty }[/math] where [math]\displaystyle{ 0\lt s\leq\frac{1}{2} }[/math], and [math]\displaystyle{ c^{2s}(H^{s}|_{E})\lt \infty }[/math], then [math]\displaystyle{ E }[/math] is rectifiable in the sense that there are countably many [math]\displaystyle{ C^{1} }[/math] curves [math]\displaystyle{ \Gamma_{i} }[/math] such that [math]\displaystyle{ H^{s}(E\backslash \bigcup\Gamma_{i})=0 }[/math]. The result is not true for [math]\displaystyle{ \frac{1}{2}\lt s\lt 1 }[/math], and [math]\displaystyle{ c^{2s}(H^{s}|_{E})=\infty }[/math] for [math]\displaystyle{ 1\lt s\leq n }[/math].:

In the opposite direction, there is a result of Peter Jones:

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

Analogous results hold in general metric spaces:

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. https://web.archive.org/web/20070821103738/http://www.lems.brown.edu/vision/people/leymarie/Notes/CurvSurf/MengerCurv/index.html. Retrieved 2007-11-19. 

References

• 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.