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.
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
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. Menger curvature can be used to give quantitative conditions for when sets in
define 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] In the opposite direction, there is a result of Peter Jones:[4] Analogous results hold in general metric spaces:[5]