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Menger curvature

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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

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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(xyz) 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(xyz) = 0. If any of the points x, y and z are coincident, again define c(xyz) = 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

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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

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

References

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  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. ^ Strzelecki, Paweł; Szumańska, Marta; von der Mosel, Heiko (2010). "Regularizing and self-avoidance effects of integral Menger curvature". Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. 9 (1): 145–187.
  3. ^ Lin, Yong; Mattila, Pertti (2000). "Menger curvature and C1 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 Academiae Scientiarum Fennicae. 32: 437–460.