Menger curvature

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

${\displaystyle c(x,y,z)={\frac {1}{R}}.}$

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

${\displaystyle c(x,y,z)={\frac {1}{R}}={\frac {4A}{|x-y||y-z||z-x|}},}$

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

Another way of computing Menger curvature is the identity

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

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

${\displaystyle c_{X}(x,y,z)=c(f(x),f(y),f(z)).}$

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.

Menger curvature can be used to give quantitative conditions for when sets in ${\displaystyle \mathbb {R} ^{n}}$ may be rectifiable. For a Borel measure ${\displaystyle \mu }$ on a Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ define

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

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

The basic intuition behind the result is that Menger curvature measures how straight a given triple of points are (the smaller ${\displaystyle c(x,y,z)\max\{|x-y|,|y-z|,|z-y|\}}$ 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 ${\displaystyle p>3}$, ${\displaystyle f:S^{1}\rightarrow \mathbb {R} ^{n}}$ be a homeomorphism and ${\displaystyle \Gamma =f(S^{1})}$. Then ${\displaystyle f\in C^{1,1-{\frac {3}{p}}}(S^{1})}$ if ${\displaystyle c^{p}(H^{1}|_{\Gamma })<\infty }$.
• If ${\displaystyle 0<H^{s}(E)<\infty }$ where ${\displaystyle 0<s\leq {\frac {1}{2}}}$, and ${\displaystyle c^{2s}(H^{s}|_{E})<\infty }$, then ${\displaystyle E}$ is rectifiable in the sense that there are countably many ${\displaystyle C^{1}}$ curves ${\displaystyle \Gamma _{i}}$ such that ${\displaystyle H^{s}(E\backslash \bigcup \Gamma _{i})=0}$. The result is not true for ${\displaystyle {\frac {1}{2}}<s<1}$, and ${\displaystyle c^{2s}(H^{s}|_{E})=\infty }$ for ${\displaystyle 1<s\leq n}$.:[3]

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

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

Analogous results hold in general metric spaces:[5]

• Menger-Melnikov curvature of a measure

• Leymarie, F. (September 2003). "Notes on Menger Curvature". Archived from the original on 2007-08-21. Retrieved 2007-11-19.

• 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.