View source for Curvature of a measure

In [[Mathematics|mathematics]], the '''curvature of a measure''' defined on the [[Euclidean plane]] '''R'''<sup>2</sup> is a quantification of how much the measure's "distribution of mass" is "curved". It is related to notions of [[Curvature|curvature]] in [[Geometry|geometry]]. In the form presented below, the concept was introduced in 1995 by the [[Biography:Mathematician|mathematician]] Mark S. Melnikov; accordingly, it may be referred to as the '''Melnikov curvature''' or '''Menger-Melnikov curvature'''. Melnikov and Verdera (1995) established a powerful connection between the curvature of measures and the Cauchy kernel.

==Definition==

Let ''μ'' be a [[Borel measure]] on the Euclidean plane '''R'''<sup>2</sup>. Given three (distinct) points ''x'', ''y'' and ''z'' in '''R'''<sup>2</sup>, let ''R''(''x'',&nbsp;''y'',&nbsp;''z'') be the [[Radius|radius]] of the Euclidean [[Circle|circle]] that joins all three of them, or +∞ if they are [[Line (geometry)|collinear]]. The [[Menger curvature]] ''c''(''x'',&nbsp;''y'',&nbsp;''z'') is defined to be

:<math>c(x, y, z) = \frac{1}{R(x, y, z)},</math>

with the natural convention that ''c''(''x'',&nbsp;''y'',&nbsp;''z'')&nbsp;=&nbsp;0 if ''x'', ''y'' and ''z'' are collinear. It is also conventional to extend this definition by setting ''c''(''x'',&nbsp;''y'',&nbsp;''z'')&nbsp;=&nbsp;0 if any of the points ''x'', ''y'' and ''z'' coincide. The '''Menger-Melnikov curvature''' ''c''<sup>2</sup>(''μ'') of ''μ'' is defined to be

:<math>c^{2} (\mu) = \iiint_{\mathbb{R}^{2}} c(x, y, z)^{2} \, \mathrm{d} \mu (x) \mathrm{d} \mu (y) \mathrm{d} \mu (z).</math>

More generally, for ''α''&nbsp;≥&nbsp;0, define ''c''<sup>2''α''</sup>(''μ'') by

:<math>c^{2 \alpha} (\mu) = \iiint_{\mathbb{R}^{2}} c(x, y, z)^{2 \alpha} \, \mathrm{d} \mu (x) \mathrm{d} \mu (y) \mathrm{d} \mu (z).</math>

One may also refer to the curvature of ''μ'' at a given point ''x'':

:<math>c^{2} (\mu; x) = \iint_{\mathbb{R}^{2}} c(x, y, z)^{2} \, \mathrm{d} \mu (y) \mathrm{d} \mu (z),</math>

in which case

:<math>c^{2} (\mu) = \int_{\mathbb{R}^{2}} c^{2} (\mu; x) \, \mathrm{d} \mu (x).</math>

==Examples==

* The [[Trivial measure|trivial measure]] has zero curvature.
* A [[Dirac measure]] ''δ''<sub>''a''</sub> supported at any point ''a'' has zero curvature.
* If ''μ'' is any measure whose [[Support (measure theory)|support]] is contained within a Euclidean line ''L'', then ''μ'' has zero curvature. For example, one-dimensional [[Lebesgue measure]] on any line (or line segment) has zero curvature.
* The Lebesgue measure defined on all of '''R'''<sup>2</sup> has infinite curvature.
* If ''μ'' is the uniform one-dimensional [[Hausdorff measure]] on a circle ''C''<sub>''r''</sub> or radius ''r'', then ''μ'' has curvature 1/''r''.

==Relationship to the Cauchy kernel==

In this section, '''R'''<sup>2</sup> is thought of as the [[Complex plane|complex plane]] '''C'''. Melnikov and Verdera (1995) showed the precise relation of the [[Bounded operator|boundedness]] of the Cauchy kernel to the curvature of measures. They proved that if there is some constant ''C''<sub>0</sub> such that

:<math>\mu(B_{r} (x)) \leq C_{0} r</math>

for all ''x'' in '''C''' and all ''r''&nbsp;&gt;&nbsp;0, then there is another constant ''C'', depending only on ''C''<sub>0</sub>, such that

:<math>\left| 6 \int_{\mathbb{C}} | \mathcal{C}_{\varepsilon} (\mu) (z) |^{2} \, \mathrm{d} \mu (z) - c_{\varepsilon}^{2} (\mu) \right| \leq C \| \mu \|</math>

for all ''ε''&nbsp;&gt;&nbsp;0. Here ''c''<sub>''ε''</sub> 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

:<math>| x - y | > \varepsilon;</math>
:<math>| y - z | > \varepsilon;</math>
:<math>| z - x | > \varepsilon.</math>

Similarly, <math>\mathcal{C}_{\varepsilon}</math> denotes a truncated Cauchy integral operator: for a measure ''μ'' on '''C''' and a point ''z'' in '''C''', define

:<math>\mathcal{C}_{\varepsilon} (\mu) (z) = \int \frac{1}{\xi - z} \, \mathrm{d} \mu (\xi),</math>

where the integral is taken over those points ''ξ'' in '''C''' with

:<math>| \xi - z | > \varepsilon.</math>

==References==

* {{cite journal
| last = Mel'nikov
| first = Mark S.
| title = Analytic capacity: a discrete approach and the curvature of measure
| journal = [[Matematicheskii Sbornik]]
| volume = 186
| year = 1995
| issue = 6
| pages = 57&ndash;76
| issn = 0368-8666
}}
* {{cite journal
|author1=Melnikov, Mark S.  |author2=Verdera, Joan
| title = A geometric proof of the ''L''<sup>2</sup> boundedness of the Cauchy integral on Lipschitz graphs
| journal =  International Mathematics Research Notices| year = 1995
| issue = 7
| pages = 325&ndash;331
| doi = 10.1155/S1073792895000249
| volume = 1995
|doi-access=free
}}
* {{cite journal
| last = Tolsa
| first = Xavier
| title = Principal values for the Cauchy integral and rectifiability
| journal = [[Organization:Proceedings of the American Mathematical Society|Proceedings of the American Mathematical Society]]
| volume = 128
| year = 2000
| issue = 7
| pages = 2111&ndash;2119
| doi = 10.1090/S0002-9939-00-05264-3
| doi-access = free
}}

[[Category:Curvature (mathematics)]]
[[Category:Measure theory]]

{{Sourceattribution|Curvature of a measure}}