my-server
← Wiki

Menger curvature

In mathematics, the Menger curvature of a triple of points in n-dimensional Euclidean space R<sup>n</sup> 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 R<sup>n</sup>; for simplicity, assume for the moment that all three points are distinct and do not lie on a single straight line. Let Π&nbsp;⊆&nbsp;R<sup>n</sup> be the Euclidean plane spanned by x, y and z and let C&nbsp;⊆&nbsp;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,&nbsp;y,&nbsp;z) 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(x,&nbsp;y,&nbsp;z)&nbsp;=&nbsp;0. If any of the points x, y and z are coincident, again define c(x,&nbsp;y,&nbsp;z)&nbsp;=&nbsp;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 {x,y,z}, and the value c<sub>X</sub> (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 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 (the smaller 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 , 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 (in particular, this measure is the Frostman measure associated to E). 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:

See also

External links

References