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| In [[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]] 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 (1995) established a powerful connection between the curvature of measures and the [[Cauchy integral formula|Cauchy kernel]].
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| ==Definition==
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| 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'', ''y'', ''z'') be the [[radius]] of the Euclidean [[circle]] that joins all three of them, or +∞ if they are [[Line (geometry)|collinear]]. The [[Menger curvature]] ''c''(''x'', ''y'', ''z'') is defined to be
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| :<math>c(x, y, z) = \frac{1}{R(x, y, z)},</math>
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| 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. The '''Menger-Melnikov curvature''' ''c''<sup>2</sup>(''μ'') of ''μ'' is defined to be
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| :<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>
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| More generally, for ''α'' ≥ 0, define ''c''<sup>2''α''</sup>(''μ'') by
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| :<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>
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| One may also refer to the curvature of ''μ'' at a given point ''x'':
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| :<math>c^{2} (\mu; x) = \iint_{\mathbb{R}^{2}} c(x, y, z)^{2} \, \mathrm{d} \mu (y) \mathrm{d} \mu (z),</math>
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| in which case
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| :<math>c^{2} (\mu) = \int_{\mathbb{R}^{2}} c^{2} (\mu; x) \, \mathrm{d} \mu (x).</math>
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| ==Examples==
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| * The [[trivial measure]] has zero curvature.
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| * A [[Dirac measure]] ''δ''<sub>''a''</sub> supported at any point ''a'' has zero curvature.
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| * 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.
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| * The Lebesgue measure defined on all of '''R'''<sup>2</sup> has infinite curvature.
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| * 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''.
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| ==Relationship to the Cauchy kernel==
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| In this section, '''R'''<sup>2</sup> is thought of as the [[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
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| :<math>\mu(B_{r} (x)) \leq C_{0} r</math>
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| for all ''x'' in '''C''' and all ''r'' > 0, then there is another constant ''C'', depending only on ''C''<sub>0</sub>, such that
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| :<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>
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| for all ''ε'' > 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
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| :<math>| x - y | > \varepsilon;</math> | |
| :<math>| y - z | > \varepsilon;</math>
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| :<math>| z - x | > \varepsilon.</math>
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| 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
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| :<math>\mathcal{C}_{\varepsilon} (\mu) (z) = \int \frac{1}{\xi - z} \, \mathrm{d} \mu (\xi),</math>
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| where the integral is taken over those points ''ξ'' in '''C''' with
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| :<math>| \xi - z | > \varepsilon.</math>
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| ==References==
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| * {{cite journal
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| | last = Mel'nikov
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| | first = Mark S.
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| | title = Analytic capacity: a discrete approach and the curvature of measure
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| | journal = [[Sbornik: Mathematics|Mat. Sb.]]
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| | volume = 186
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| | year = 1995
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| | issue = 6
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| | pages = 57–76
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| | issn = 0368-8666
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| }}
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| * {{cite journal
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| | author = Melnikov, Mark S. and Verdera, Joan
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| | title = A geometric proof of the ''L''<sup>2</sup> boundedness of the Cauchy integral on Lipschitz graphs
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| | journal = Internat. Math. Res. Notices
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| | year = 1995
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| | issue = 7
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| | pages = 325–331
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| | doi = 10.1155/S1073792895000249
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| | volume = 1995
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| }}
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| * {{cite journal
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| | last = Tolsa
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| | first = Xavier
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| | title = Principal values for the Cauchy integral and rectifiability
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| | journal = [[Proceedings of the American Mathematical Society|Proc. Amer. Math. Soc.]]
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| | volume = 128
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| | year = 2000
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| | issue = 7
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| | pages = 2111–2119
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| | doi = 10.1090/S0002-9939-00-05264-3
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| }}
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| [[Category:Curvature (mathematics)]]
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| [[Category:Measure theory]]
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