Curvature of a measure

Melnikov and Verdera (1995) established a powerful connection between the curvature of measures and the Cauchy kernel.

Given three (distinct) points x, y and z in R2, let R(x, y, z) be the radius of the Euclidean circle that joins all three of them, or +∞ if they are collinear.

The Menger curvature c(x, y, z) is defined to be with the natural convention that c(x, y, z) = 0 if x, y and z are collinear.

It is also conventional to extend this definition by setting c(x, y, z) = 0 if any of the points x, y and z coincide.

Here cε denotes a truncated version of the Menger-Melnikov curvature in which the integral is taken only over those points x, y and z such that Similarly,