# Brascamp-Lieb inequality

In mathematics, the Brascamp-Lieb inequality is a result in geometry concerning integrable functions on "n"-dimensional Euclidean space R"n". It generalizes the Loomis-Whitney inequality, the Prékopa-Leindler inequality and Hölder's inequality, and is named after Herm Jan Brascamp and Elliott H. Lieb. The original inequality (called the geometric inequality here) is in [H.J. Brascamp and E.H. Lieb, "Best Constants in Young's Inequality, Its""Converse and Its Generalization to More Than Three Functions", Adv. in Math.20, 151-172 (1976).] .Its generalization, stated first, is in [E.H.Lieb, "Gaussian Kernels have only Gaussian Maximizers", Inventiones Mathematicae102, pp. 179-208 (1990).]

tatement of the inequality

Fix natural numbers "m" and "n". For 1 &le; "i" &le; "m", let "n""i" &isin; N and let "c""i" &gt; 0 so that

:$sum_\left\{i = 1\right\}^\left\{m\right\} c_\left\{i\right\} n_\left\{i\right\} = n.$

Choose non-negative, integrable functions

:$f_\left\{i\right\} in L^\left\{1\right\} left\left( mathbb\left\{R\right\}^\left\{n_\left\{i ; \left[0, + infty\right] ight\right)$

and surjective linear maps

:$B_\left\{i\right\} : mathbb\left\{R\right\}^\left\{n\right\} o mathbb\left\{R\right\}^\left\{n_\left\{i.$

Then the following inequality holds:

:$int_\left\{mathbb\left\{R\right\}^\left\{n prod_\left\{i = 1\right\}^\left\{m\right\} f_\left\{i\right\} left\left( B_\left\{i\right\} x ight\right)^\left\{c_\left\{i , mathrm\left\{d\right\} x leq D^\left\{- 1/2\right\} prod_\left\{i = 1\right\}^\left\{m\right\} left\left( int_\left\{mathbb\left\{R\right\}^\left\{n_\left\{i\right\} f_\left\{i\right\} \left(y\right) , mathrm\left\{d\right\} y ight\right)^\left\{c_\left\{i,$

where "D" is given by

:$D = inf left\left\{ left. frac\left\{det left\left( sum_\left\{i = 1\right\}^\left\{m\right\} c_\left\{i\right\} B_\left\{i\right\}^\left\{*\right\} A_\left\{i\right\} B_\left\{i\right\} ight\right)\right\}\left\{prod_\left\{i = 1\right\}^\left\{m\right\} \left( det A_\left\{i\right\} \right)^\left\{c_\left\{i\right\} ight| A_\left\{i\right\} mbox\left\{ is a positive-definite \right\} n_\left\{i\right\} imes n_\left\{i\right\} mbox\left\{ matrix\right\} ight\right\}.$

Another way to state this is that the constant "D" is what one would obtain byrestricting attention to the case in which each $f_\left\{i\right\}$ is a centered Gaussianfunction, namely $f_\left\{i\right\}\left(y\right) = exp \left\{-\left(y,, A_\left\{i\right\}, y\right)\right\}$.

Relationships to other inequalities

The geometric Brascamp-Lieb inequality

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.

For "i" = 1, ..., "m", let "c""i" &gt; 0 and let "u""i" &isin; S"n"−1 be a unit vector; suppose that that "c""i" and "u""i" satisfy

:$x = sum_\left\{i = 1\right\}^\left\{m\right\} c_\left\{i\right\} \left(x cdot u_\left\{i\right\}\right) u_\left\{i\right\}$

for all "x" in R"n". Let "f""i" &isin; "L"1(R; [0, +&infin;] ) for each "i" = 1, ..., "m". Then

:$int_\left\{mathbb\left\{R\right\}^\left\{n prod_\left\{i = 1\right\}^\left\{m\right\} f_\left\{i\right\} \left(x cdot u_\left\{i\right\}\right)^\left\{c_\left\{i , mathrm\left\{d\right\} x leq prod_\left\{i = 1\right\}^\left\{m\right\} left\left( int_\left\{mathbb\left\{R f_\left\{i\right\} \left(y\right) , mathrm\left\{d\right\} y ight\right)^\left\{c_\left\{i.$

The geometric Brascamp-Lieb inequality follows from the Brascamp-Lieb inequality as stated above by taking "n""i" = 1 and "B""i"("x") = "x" &middot; "u""i". Then, for "z""i" &isin; R,

:$B_\left\{i\right\}^\left\{*\right\} \left(z_\left\{i\right\}\right) = z_\left\{i\right\} u_\left\{i\right\}.$

It follows that "D" = 1 in this case.

Hölder's inequality

As another special case, take "n""i" = "n", "B""i" = id, the identity map on R"n", replacing "f""i" by $f_\left\{i\right\}^\left\{1/c_\left\{i$, and let "c""i" = 1 / "p""i" for 1 &le; "i" &le; "m". Then

:$sum_\left\{i = 1\right\}^\left\{m\right\} p_\left\{i\right\} = 1$

and the log-concavity of the determinant of a positive definite matrix implies that "D" = 1. This yields Hölder's inequality in R"n":

:$int_\left\{mathbb\left\{R\right\}^\left\{n prod_\left\{i = 1\right\}^\left\{m\right\} f_\left\{i\right\} \left(x\right) , mathrm\left\{d\right\} x leq prod_\left\{i = 1\right\}^\left\{m\right\} | f_\left\{i\right\} |_\left\{p_\left\{i.$

References

* cite book
last = Ball
first = Keith M.
chapter = Volumes of sections of cubes and related problems
title = Geometric aspects of functional analysis (1987--88)
editor = J. Lindenstrauss and V.D. Milman
series = Lecture Notes in Math., Vol. 1376
pages = pp. 251&ndash;260
publisher = Springer
location = Berlin
year = 1989

* cite journal
last=Gardner
first=Richard J.
title=The Brunn-Minkowski inequality
journal=Bull. Amer. Math. Soc. (N.S.)
volume=39
issue=3
year=2002
pages= pp. 355&ndash;405 (electronic)
issn = 0273-0979
url = http://www.ams.org/bull/2002-39-03/S0273-0979-02-00941-2/S0273-0979-02-00941-2.pdf
doi=10.1090/S0273-0979-02-00941-2

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