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 cX (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 $$\mu$$ 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 $$ H^{1}|_{E} $$ denotes one-dimensional Hausdorff measure restricted to the set $$ E$$.

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


 * Let $$ p>3$$, $$ f:S^{1}\rightarrow \mathbb{R}^{n}$$ be a homeomorphism and $$\Gamma=f(S^{1})$$. Then $$ f\in C^{1,1-\frac{3}{p}}(S^{1})$$ if $$ c^{p}(H^{1}|_{\Gamma})<\infty$$.
 * If $$ 00$$, 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 $$c^{2}(\mu)<\infty$$ (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: