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

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

where $\mathrm{\angle }xyz$ 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 ${ℝ}^2$. 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 ${ℝ}^n$ may be rectifiable. For a Borel measure $\mu$ on a Euclidean space ${ℝ}^n$ define

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

• A Borel set $E\subseteq {ℝ}^n$ is rectifiable if $c^2(H^1{|}_E)<\mathrm{\infty }$, 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 $p>3$, $f:S^1\to {ℝ}^n$ be a homeomorphism and $\mathrm{\Gamma }=f(S^1)$. Then $f\in C^{1,1-\frac{3}{p}}(S^1)$ if $c^p(H^1{|}_{\mathrm{\Gamma }})<\mathrm{\infty }$.
• If $0<H^s(E)<\mathrm{\infty }$ where $0<s\le \frac{1}{2}$, and $c^{2s}(H^s{|}_E)<\mathrm{\infty }$, then $E$ is rectifiable in the sense that there are countably many $C^1$ curves ${\mathrm{\Gamma }}_i$ such that $H^s(E\mathrm{\setminus }\bigcup {\mathrm{\Gamma }}_i)=0$. The result is not true for $\frac{1}{2}<s<1$, and $c^{2s}(H^s{|}_E)=\mathrm{\infty }$ for $1<s\le n$.:

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

• If $E\subseteq \mathrm{\Gamma }\subseteq {ℝ}^2$, $H^1(E)>0$, and $\mathrm{\Gamma }$ is rectifiable. Then there is a positive Radon measure $\mu$ supported on $E$ satisfying $\mu B(x,r)\le r$ for all $x\in E$ and $r>0$ such that $c^2(\mu )<\mathrm{\infty }$ (in particular, this measure is the Frostman measure associated to E). Moreover, if $H^1(B(x,r)\cap \mathrm{\Gamma })\le Cr$ for some constant C and all $x\in \mathrm{\Gamma }$ and r>0, then $c^2(H^1{|}_E)<\mathrm{\infty }$. 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

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References

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