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| In [[mathematics]], the '''Borell–Brascamp–Lieb inequality''' is an [[integral]] [[inequality (mathematics)|inequality]] due to many different mathematicians but named after [[Christer Borell]], [[Herm Jan Brascamp]] and [[Elliott Lieb]].
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| The result was proved for ''p'' > 0 by Henstock and Macbeath in 1953. The case ''p'' = 0 is known as the [[Prékopa–Leindler inequality]] and was re-discovered by Brascamp and Lieb in 1976, when they proved the general version below; working independently, Borell had done the same in 1975. The nomenclature of "Borell–Brascamp–Lieb inequality" is due to Cordero-Erausquin, McCann and Schmuckenschläger, who in 2001 generalized the result to [[Riemannian manifold]]s such as the [[sphere]] and [[hyperbolic space]].
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| ==Statement of the inequality in '''R'''<sup>''n''</sup>==
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| Let 0 < ''λ'' < 1, let −1 / ''n'' ≤ ''p'' ≤ +∞, and let ''f'', ''g'', ''h'' : '''R'''<sup>''n''</sup> → [0, +∞) be integrable functions such that, for all ''x'' and ''y'' in '''R'''<sup>''n''</sup>,
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| :<math>h \left( (1 - \lambda) x + \lambda y \right) \geq M_{p} \left( f(x), g(y), \lambda \right),</math>
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| where
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| :<math>
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| \begin{align}
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| M_{p} (a, b, \lambda)& = \left( (1 - \lambda) a^{p} + \lambda b^{p} \right)^{1/p},\\
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| M_{0} (a, b, \lambda)& = a^{1 - \lambda} b^{\lambda}.\,
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| \end{align}
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| </math> | |
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| Then
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| :<math>\int_{\mathbb{R}^{n}} h(x) \, \mathrm{d} x \geq M_{p / (n p + 1)} \left( \int_{\mathbb{R}^{n}} f(x) \, \mathrm{d} x, \int_{\mathbb{R}^{n}} g(x) \, \mathrm{d} x, \lambda \right).</math>
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| (When ''p'' = −1 / ''n'', the convention is to take ''p'' / (''n'' ''p'' + 1) to be −∞; when ''p'' = +∞, it is taken to be 1 / ''n''.)
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| ==References==
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| * {{cite journal
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| | last = Borell
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| | first = Christer
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| | title = Convex set functions in ''d''-space
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| | journal = Period. Math. Hungar.
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| | volume = 6
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| | year = 1975
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| | number = 2
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| | pages = 111–136
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| | doi = 10.1007/BF02018814
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| }}
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| * {{cite journal
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| | author = Brascamp, Herm Jan and Lieb, Elliott H.
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| | title = On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation
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| | journal = J. Functional Analysis
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| | volume = 22
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| | year = 1976
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| | number = 4
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| | pages = 366–389
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| | doi = 10.1016/0022-1236(76)90004-5
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| }}
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| * {{cite journal
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| | author = Cordero-Erausquin, Dario, McCann, Robert J. and Schmuckenschläger, Michael
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| | title = A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
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| | journal = Invent. Math.
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| | volume = 146
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| | year = 2001
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| | number = 2
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| | pages = 219–257
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| | doi = 10.1007/s002220100160
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| }}
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| * {{cite journal
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| | last=Gardner
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| | first=Richard J.
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| | title=The Brunn–Minkowski inequality
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| | journal=[[Bulletin of the American Mathematical Society|Bull. Amer. Math. Soc.]] (N.S.)
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| | volume=39
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| | issue=3
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| | year=2002
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| | pages=355–405 (electronic)
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| | url = http://www.ams.org/bull/2002-39-03/S0273-0979-02-00941-2/S0273-0979-02-00941-2.pdf
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| | doi=10.1090/S0273-0979-02-00941-2
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| }}
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| * {{cite journal
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| | author = Henstock, R. and [[Alexander M. Macbeath|Macbeath, A. M.]]
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| | title = On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik
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| | journal = Proc. London Math. Soc. (3)
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| | volume = 3
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| | year = 1953
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| | pages = 182–194
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| | doi = 10.1112/plms/s3-3.1.182
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| }}
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| {{DEFAULTSORT:Borell-Brascamp-Lieb inequality}}
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| [[Category:Geometric inequalities]]
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| [[Category:Integral geometry]]
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I would like to introduce myself to you, I am Jayson Simcox but I don't like when individuals use my full title. I've always loved living in Alaska. What I love doing is soccer but I don't have the time lately. Distributing manufacturing has been his occupation for some time.
Feel free to visit my blog ... free psychic readings