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. Let R be the radius of C. Then the Menger curvature c of x, y and z is defined by
If the three points are collinear, R can be informally considered to be +∞, and it makes rigorous sense to define c = 0. If any of the points x, y and z are coincident, again define c = 0.
Using the well-known formula relating the side lengths of a triangle to its area, it follows that
where A denotes the area of the triangle spanned by x, y and z.
Another way of computing Menger curvature is the identity
where 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 into. Define the Menger curvature of these points to be
Note that f need not be defined on all of X, just on ', and the value c_X ' is independent of the choice of f.

Integral Curvature Rectifiability

Menger curvature can be used to give quantitative conditions for when sets in may be rectifiable. For a Borel measure on a Euclidean space define

A Borel set is rectifiable if, where denotes one-dimensional Hausdorff measure restricted to the set.

The basic intuition behind the result is that Menger curvature measures how straight a given triple of points are, 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, be a homeomorphism and. Then if.
If where, and, then is rectifiable in the sense that there are countably many curves such that. The result is not true for, and for.:

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

If,, and is rectifiable. Then there is a positive Radon measure supported on satisfying for all and such that . Moreover, if for some constant C and all and r>0, then. This last result follows from the Analyst's Traveling Salesman Theorem.

Analogous results hold in general metric spaces: