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| In [[mathematics]], the '''Prékopa–Leindler inequality''' is an [[integral]] [[inequality (mathematics)|inequality]] closely related to the [[reverse Young's inequality]], the [[Brunn–Minkowski inequality]] and a number of other important and classical inequalities in [[mathematical analysis|analysis]]. The result is named after the [[Hungary|Hungarian]] [[mathematician]]s [[András Prékopa]] and [[László Leindler]].
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| ==Statement of the inequality==
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| Let 0 < ''λ'' < 1 and let ''f'', ''g'', ''h'' : '''R'''<sup>''n''</sup> → [0, +∞) be non-[[negative number|negative]] [[real number|real-valued]] [[measurable function]]s defined on ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup>. Suppose that these functions satisfy
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| {{NumBlk|:|<math>h \left( (1-\lambda)x + \lambda y \right) \geq f(x)^{1 - \lambda} g(y)^{\lambda}</math>|{{EquationRef|1}}}}
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| for all ''x'' and ''y'' in '''R'''<sup>''n''</sup>. Then
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| :<math>\| h\|_{1} := \int_{\mathbb{R}^{n}} h(x) \, \mathrm{d} x \geq \left( \int_{\mathbb{R}^{n}} f(x) \, \mathrm{d} x \right)^{1 -\lambda} \left( \int_{\mathbb{R}^{n}} g(x) \, \mathrm{d} x \right)^{\lambda} =: \| f\|_{1}^{1 -\lambda} \| g\|_{1}^{\lambda}. \, </math>
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| ==Essential form of the inequality==
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| Recall that the [[essential supremum]] of a measurable function ''f'' : '''R'''<sup>''n''</sup> → '''R''' is defined by
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| :<math>\mathop{\mathrm{ess\,sup}}_{x \in \mathbb{R}^{n}} f(x) = \inf \left\{ t \in [- \infty, + \infty] | f(x) \leq t \mbox{ for almost all } x \in \mathbb{R}^{n} \right\}.</math>
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| This notation allows the following ''essential form'' of the Prékopa–Leindler inequality: let 0 < ''λ'' < 1 and let ''f'', ''g'' ∈ ''L''<sup>1</sup>('''R'''<sup>''n''</sup>; [0, +∞)) be non-negative [[absolutely integrable]] functions. Let
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| :<math>s(x) = \mathop{\mathrm{ess\,sup}}_{y \in \mathbb{R}^{n}} f \left( \frac{x - y}{1 - \lambda} \right)^{1 - \lambda} g \left( \frac{y}{\lambda} \right)^{\lambda}.</math>
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| Then ''s'' is measurable and
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| :<math>\| s \|_{1} \geq \| f \|_{1}^{1 - \lambda} \| g \|_{1}^{\lambda}.</math>
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| The essential supremum form was given in.<ref>{{cite journal | authors = [[Herm Jan Brascamp]] and [[Elliott H. Lieb]] | title = On extensions of the Brunn–Minkowski and Prekopa–Leindler theorems, including inequalities for log concave functions and with an application to the diffusion equation | journal = Journal of Functional Analysis | volume = 22 |issue=4 | pages = 366–389 | year = 1976 |doi=10.1016/0022-1236(76)90004-5 }}</ref> Its use can change the left side of the inequality. For example, a function ''g'' that takes the value 1 at exactly one point will not usually yield a zero left side in the "non-essential sup" form but it will always yield a zero left side in the "essential sup" form.
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| ==Relationship to the Brunn–Minkowski inequality==
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| It can be shown that the usual Prékopa–Leindler inequality implies the [[Brunn–Minkowski inequality]] in the following form: if 0 < ''λ'' < 1 and ''A'' and ''B'' are [[bounded set|bounded]], [[measurable set|measurable subsets]] of '''R'''<sup>''n''</sup> such that the [[Minkowski sum]] (1 − ''λ'')''A'' + λ''B'' is also measurable, then
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| :<math>\mu \left( (1 - \lambda) A + \lambda B \right) \geq \mu (A)^{1 - \lambda} \mu (B)^{\lambda},</math>
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| where ''μ'' denotes ''n''-dimensional [[Lebesgue measure]]. Hence, the Prékopa–Leindler inequality can also be used<ref>Gardner, Richard J. (2002). "The Brunn–Minkowski inequality". Bull. Amer. Math. Soc. (N.S.) 39 (3): pp. 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2. ISSN 0273-0979.</ref> to prove the Brunn–Minkowski inequality in its more familiar form: if 0 < ''λ'' < 1 and ''A'' and ''B'' are non-[[empty set|empty]], [[bounded set|bounded]], [[measurable set|measurable subsets]] of '''R'''<sup>''n''</sup> such that (1 − ''λ'')''A'' + λ''B'' is also measurable, then
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| :<math>\mu \left( (1 - \lambda) A + \lambda B \right)^{1 / n} \geq (1 - \lambda) \mu (A)^{1 / n} + \lambda \mu (B)^{1 / n}.</math>
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| ==Applications in probability and statistics==
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| The Prékopa–Leindler inequality is useful in the theory of [[Logarithmically concave function|log-concave distributions]], as it can be used to show that log-concavity is preserved by [[Marginal distribution|marginalization]] and [[Independence (probability)|independent]] summation of log-concave distributed random variables. Suppose that ''H(x,y)'' is a log-concave distribution for ''(x,y)'' ∈ '''R'''<sup>''m''</sup> × '''R'''<sup>''n''</sup>, so that by definition we have
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| {{NumBlk|:|<math>H \left( (1 - \lambda)(x_1,y_1) + \lambda (x_2,y_2) \right) \geq H(x_1,y_1)^{1 - \lambda} H(x_2,y_2)^{\lambda},</math>|{{EquationRef|2}}}}
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| and let ''M(y)'' denote the marginal distribution obtained by integrating over ''x'':
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| :<math>M(y) = \int_{\mathbb{R}^{m}} H(x,y) dx.</math>
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| Let ''y<sub>1</sub>, y<sub>2</sub>'' ∈ '''R'''<sup>''n''</sup> and 0 < ''λ'' < 1 be given. Then equation ({{EquationNote|2}}) satisfies condition ({{EquationNote|1}}) with ''h(x) = H(x,(1-λ)y<sub>1</sub> + λy<sub>2</sub>)'', ''f(x)'' = ''H(x,y<sub>1</sub>)'' and ''g(x)'' = ''H(x,y<sub>2</sub>)'', so the Prékopa–Leindler inequality applies. It can be written in terms of ''M'' as
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| :<math>M((1-\lambda) y_1 + \lambda y_2) \geq M(y_1)^{1-\lambda} M(y_2)^\lambda,</math>
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| which is the definition of log-concavity for ''M''.
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| To see how this implies the preservation of log-convexity by independent sums, suppose that ''X'' and ''Y'' are independent random variables with log-concave distribution. Since the product of two log-concave functions is log-concave, the joint distribution of ''(X,Y)'' is also log-concave. Log-concavity is preserved by affine changes of coordinates, so the distribution of ''(X+Y,X-Y)'' is log-concave as well. Since the distribution of ''X+Y'' is a marginal over the joint distribution of ''(X+Y,X-Y)'', we conclude that ''X+Y'' has a log-concave distribution.
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| ==Notes==
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| <references/>
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| ==References==
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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 = pp. 355–405 (electronic)
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| | issn = 0273-0979
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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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| | last=Prékopa
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| | first=András
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| | title=Logarithmic concave measures with application to stochastic programming
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| | journal=[[Acta Scientiarum Mathematicarum|Acta Sci. Math.]]
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| | volume=32
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| | year=1971
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| | pages=pp. 301–316
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| | url=http://rutcor.rutgers.edu/~prekopa/SCIENT1.pdf
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| }}
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| * {{ cite journal
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| | last=Prékopa
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| | first=András
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| | title=On logarithmic concave measures and functions
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| | journal=[[Acta Scientiarum Mathematicarum|Acta Sci. Math.]]
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| | volume=34
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| | year=1973
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| | pages=pp. 335–343
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| | url=http://rutcor.rutgers.edu/~prekopa/SCIENT2.pdf
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| }}
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| {{DEFAULTSORT:Prekopa-Leindler Inequality}}
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| [[Category:Geometric inequalities]]
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| [[Category:Integral geometry]]
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| [[Category:Real analysis]]
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| [[Category:Theorems in analysis]]
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