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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.

Contents

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 ) = 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 ) = 1 R = 4 A | 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 ) = 2 sin ⁡ ∠ x y z | x − z |

where ∠ x y z 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 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 R n may be rectifiable. For a Borel measure μ on a Euclidean space R n define

c p ( μ ) = ∫ ∫ ∫ c ( x , y , z ) p d μ ( x ) d μ ( y ) d μ ( z ) .
  • A Borel set E ⊆ R n is rectifiable if c 2 ( H 1 | E ) < ∞ , 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 → R n be a homeomorphism and Γ = f ( S 1 ) . Then f ∈ C 1 , 1 − 3 p ( S 1 ) if c p ( H 1 | Γ ) < ∞ .
  • If 0 < H s ( E ) < ∞ where 0 < s ≤ 1 2 , and c 2 s ( H s | E ) < ∞ , then E is rectifiable in the sense that there are countably many C 1 curves Γ i such that H s ( E ∖ ⋃ Γ i ) = 0 . The result is not true for 1 2 < s < 1 , and c 2 s ( H s | E ) = ∞ for 1 < s ≤ n .:

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

  • If E ⊆ Γ ⊆ R 2 , H 1 ( E ) > 0 , and Γ is rectifiable. Then there is a positive Radon measure μ supported on E satisfying μ B ( x , r ) ≤ r for all x ∈ E and r > 0 such that c 2 ( μ ) < ∞ (in particular, this measure is the Frostman measure associated to E). Moreover, if H 1 ( B ( x , r ) ∩ Γ ) ≤ C r for some constant C and all x ∈ Γ and r>0, then c 2 ( H 1 | E ) < ∞ . This last result follows from the Analyst's Traveling Salesman Theorem.

    • Analogous results hold in general metric spaces:

    References

    Menger curvature Wikipedia
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