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| In [[mathematics]], the '''Brascamp–Lieb inequality''' is a result in [[geometry]] concerning [[integrable function]]s on ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup>. It generalizes the [[Loomis–Whitney inequality]] and [[Hölder's inequality]], and is named after [[Herm Jan Brascamp]] and [[Elliott H. Lieb]].
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| The original inequality (called the geometric inequality here) is in
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| .<ref>H.J. Brascamp and E.H. Lieb,
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| ''Best Constants in Young's Inequality, Its''
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| ''Converse and Its Generalization to More Than Three Functions'', Adv. in Math.
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| '''20''', 151–172 (1976).</ref>
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| Its generalization, stated first, is in
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| <ref>E.H.Lieb, ''Gaussian Kernels have only Gaussian Maximizers'', Inventiones Mathematicae '''102''', pp. 179–208 (1990).</ref>
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| ==Statement of the inequality==
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| Fix [[natural number]]s ''m'' and ''n''. For 1 ≤ ''i'' ≤ ''m'', let ''n''<sub>''i''</sub> ∈ '''N''' and let ''c''<sub>''i''</sub> > 0 so that
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| :<math>\sum_{i = 1}^{m} c_{i} n_{i} = n.</math>
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| Choose non-negative, integrable functions
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| :<math>f_{i} \in L^{1} \left( \mathbb{R}^{n_{i}} ; [0, + \infty] \right)</math>
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| and [[surjective]] [[linear map]]s
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| :<math>B_{i} : \mathbb{R}^{n} \to \mathbb{R}^{n_{i}}.</math>
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| Then the following inequality holds:
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| :<math>\int_{\mathbb{R}^{n}} \prod_{i = 1}^{m} f_{i} \left( B_{i} x \right)^{c_{i}} \, \mathrm{d} x \leq D^{- 1/2} \prod_{i = 1}^{m} \left( \int_{\mathbb{R}^{n_{i}}} f_{i} (y) \, \mathrm{d} y \right)^{c_{i}},</math>
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| where ''D'' is given by | |
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| :<math>D = \inf \left\{ \left. \frac{\det \left( \sum_{i = 1}^{m} c_{i} B_{i}^{*} A_{i} B_{i} \right)}{\prod_{i = 1}^{m} ( \det A_{i} )^{c_{i}}} \right| A_{i} \mbox{ is a positive-definite } n_{i} \times n_{i} \mbox{ matrix} \right\}.</math>
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| Another way to state this is that the constant ''D'' is what one would obtain by
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| restricting attention to the case in which each <math>f_{i}</math> is a centered Gaussian
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| function, namely <math>f_{i}(y) = \exp \{-(y,\, A_{i}\, y)\}.</math>
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| ==Relationships to other inequalities==
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| ===The geometric Brascamp–Lieb inequality===
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| The geometric Brascamp–Lieb inequality is a special case of the above, and was used by Ball (1989) to provide upper bounds for volumes of central sections of cubes.
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| For ''i'' = 1, ..., ''m'', let ''c''<sub>''i''</sub> > 0 and let ''u''<sub>''i''</sub> ∈ '''S'''<sup>''n''−1</sup> be a unit vector; suppose that that ''c''<sub>''i''</sub> and ''u''<sub>''i''</sub> satisfy
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| :<math>x = \sum_{i = 1}^{m} c_{i} (x \cdot u_{i}) u_{i}</math>
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| for all ''x'' in '''R'''<sup>''n''</sup>. Let ''f''<sub>''i''</sub> ∈ ''L''<sup>1</sup>('''R'''; [0, +∞]) for each ''i'' = 1, ..., ''m''. Then
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| :<math>\int_{\mathbb{R}^{n}} \prod_{i = 1}^{m} f_{i} (x \cdot u_{i})^{c_{i}} \, \mathrm{d} x \leq \prod_{i = 1}^{m} \left( \int_{\mathbb{R}} f_{i} (y) \, \mathrm{d} y \right)^{c_{i}}.</math>
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| The geometric Brascamp–Lieb inequality follows from the Brascamp–Lieb inequality as stated above by taking ''n''<sub>''i''</sub> = 1 and ''B''<sub>''i''</sub>(''x'') = ''x'' · ''u''<sub>''i''</sub>. Then, for ''z''<sub>''i''</sub> ∈ '''R''',
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| :<math>B_{i}^{*} (z_{i}) = z_{i} u_{i}.</math>
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| It follows that ''D'' = 1 in this case.
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| ===Hölder's inequality===
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| As another special case, take ''n''<sub>''i''</sub> = ''n'', ''B''<sub>''i''</sub> = id, the [[identity function|identity map]] on '''R'''<sup>''n''</sup>, replacing ''f''<sub>''i''</sub> by ''f''{{su|b=''i''|p=1/''c''<sub>''i''</sub>}}, and let ''c''<sub>''i''</sub> = 1 / ''p''<sub>''i''</sub> for 1 ≤ ''i'' ≤ ''m''. Then
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| :<math>\sum_{i = 1}^{m} \frac{1}{p_{i}} = 1</math>
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| and the [[Logarithmically concave function|log-concavity]] of the [[determinant]] of a [[positive definite matrix]] implies that ''D'' = 1. This yields Hölder's inequality in '''R'''<sup>''n''</sup>:
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| :<math>\int_{\mathbb{R}^{n}} \prod_{i = 1}^{m} f_{i} (x) \, \mathrm{d} x \leq \prod_{i = 1}^{m} \| f_{i} \|_{p_{i}}.</math>
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| ==References==
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| <references/> | |
| * {{cite book
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| | last = Ball
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| | first = Keith M.
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| | chapter = Volumes of sections of cubes and related problems
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| | title = Geometric aspects of functional analysis (1987–88)
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| | editor = [[Joram Lindenstrauss|J. Lindenstrauss]] and V.D. Milman
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| | series = Lecture Notes in Math., Vol. 1376
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| | pages = 251–260
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| |publisher = Springer
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| | location = Berlin
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| | year = 1989
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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= pp. 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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| {{DEFAULTSORT:Brascamp-Lieb Inequality}}
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| [[Category:Geometric inequalities]]
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