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

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

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 cX (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 .[1]

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[2]

  • 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 .:[3]

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

  • 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:[5]

See also

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

References

  1. Leger, J. (1999). "Menger curvature and rectifiability" (PDF). Annals of Mathematics. 149 (3): 831–869. arXiv:math/9905212. doi:10.2307/121074. JSTOR 121074. S2CID 216176.
  2. Pawl Strzelecki; Marta Szumanska; Heiko von der Mosel. "Regularizing and self-avoidance effects of integral Menger curvature". Institut F¨ur Mathematik.
  3. Yong Lin and Pertti Mattila (2000). "Menger curvature and regularity of fractals" (PDF). Proceedings of the American Mathematical Society. 129 (6): 1755–1762. doi:10.1090/s0002-9939-00-05814-7.
  4. Pajot, H. (2000). Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral. Springer. ISBN 3-540-00001-1.
  5. Schul, Raanan (2007). "Ahlfors-regular curves in metric spaces" (PDF). Annales Academiæ Scientiarum Fennicæ. 32: 437–460.
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