Curvature of a measure


In mathematics, the curvature of a measure defined on the Euclidean plane R2 is a quantification of how much the measure's "distribution of mass" is "curved". It is related to notions of curvature in geometry. In the form presented below, the concept was introduced in 1995 by the mathematician Mark S. Melnikov; accordingly, it may be referred to as the Melnikov curvature or Menger-Melnikov curvature. Melnikov and Verdera established a powerful connection between the curvature of measures and the Cauchy kernel.

Definition

Let μ be a Borel measure on the Euclidean plane R2. Given three points x, y and z in R2, let R be the radius of the Euclidean circle that joins all three of them, or +∞ if they are collinear. The Menger curvature c is defined to be
with the natural convention that c = 0 if x, y and z are collinear. It is also conventional to extend this definition by setting c = 0 if any of the points x, y and z coincide. The Menger-Melnikov curvature c2 of μ is defined to be
More generally, for α ≥ 0, define c2α by
One may also refer to the curvature of μ at a given point x:
in which case

Examples

In this section, R2 is thought of as the complex plane C. Melnikov and Verdera showed the precise relation of the boundedness of the Cauchy kernel to the curvature of measures. They proved that if there is some constant C0 such that
for all x in C and all r > 0, then there is another constant C, depending only on C0, such that
for all ε > 0. 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, denotes a truncated Cauchy integral operator: for a measure μ on C and a point z in C, define
where the integral is taken over those points ξ in C with