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)={\frac {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)={\frac {1}{R}}={\frac {4A}{|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 $\mu$ 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 $E$ .^[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 $\Gamma =f(S^{1})$ . 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 $C^{1}$ curves $\Gamma _{i}$ 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 $E\subseteq \Gamma \subseteq \mathbb {R} ^{2}$ , $H^{1}(E)>0$ , 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 $c^{2}(\mu )<\infty$ (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.