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 ≤ "i" ≤ "m", let "n"_"i" ∈ N and let "c"_"i" > 0 so that

:$sum\_\{i\; =\; 1\}^\{m\}\; c\_\{i\}\; n\_\{i\}\; =\; n.$

Choose non-negative, integrable functions

:$f\_\{i\}\; in\; L^\{1\}\; left(\; mathbb\{R\}^\{n\_\{i\; ;\; [0,\; +\; infty]\; ight)$

and surjective linear maps

:$B\_\{i\}\; :\; mathbb\{R\}^\{n\}\; o\; mathbb\{R\}^\{n\_\{i.$

Then the following inequality holds:

:$int\_\{mathbb\{R\}^\{n\; prod\_\{i\; =\; 1\}^\{m\}\; f\_\{i\}\; left(\; B\_\{i\}\; x\; ight)^\{c\_\{i\; ,\; mathrm\{d\}\; x\; leq\; D^\{-\; 1/2\}\; prod\_\{i\; =\; 1\}^\{m\}\; left(\; int\_\{mathbb\{R\}^\{n\_\{i\}\; f\_\{i\}\; (y)\; ,\; mathrm\{d\}\; y\; ight)^\{c\_\{i,$

where "D" is given by

:$D\; =\; inf\; left\{\; left.\; frac\{det\; left(\; sum\_\{i\; =\; 1\}^\{m\}\; c\_\{i\}\; B\_\{i\}^\{*\}\; A\_\{i\}\; B\_\{i\}\; ight)\}\{prod\_\{i\; =\; 1\}^\{m\}\; (\; det\; A\_\{i\}\; )^\{c\_\{i\}\; ight|\; A\_\{i\}\; mbox\{\; is\; a\; positive-definite\; \}\; n\_\{i\}\; imes\; n\_\{i\}\; mbox\{\; matrix\}\; ight\}.$

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

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" > 0 and let "u"_"i" ∈ S^"n"−1 be a unit vector; suppose that that "c"_"i" and "u"_"i" satisfy

:$x\; =\; sum\_\{i\; =\; 1\}^\{m\}\; c\_\{i\}\; (x\; cdot\; u\_\{i\})\; u\_\{i\}$

for all "x" in R^"n". Let "f"_"i" ∈ "L"¹(R; [0, +∞] ) for each "i" = 1, ..., "m". Then

:$int\_\{mathbb\{R\}^\{n\; prod\_\{i\; =\; 1\}^\{m\}\; f\_\{i\}\; (x\; cdot\; u\_\{i\})^\{c\_\{i\; ,\; mathrm\{d\}\; x\; leq\; prod\_\{i\; =\; 1\}^\{m\}\; left(\; int\_\{mathbb\{R\; f\_\{i\}\; (y)\; ,\; mathrm\{d\}\; y\; ight)^\{c\_\{i.$

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

:$B\_\{i\}^\{*\}\; (z\_\{i\})\; =\; z\_\{i\}\; u\_\{i\}.$

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 $f\_\{i\}^\{1/c\_\{i$, and let "c"_"i" = 1 / "p"_"i" for 1 ≤ "i" ≤ "m". Then

:$sum\_\{i\; =\; 1\}^\{m\}\; p\_\{i\}\; =\; 1$

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

:$int\_\{mathbb\{R\}^\{n\; prod\_\{i\; =\; 1\}^\{m\}\; f\_\{i\}\; (x)\; ,\; mathrm\{d\}\; x\; leq\; prod\_\{i\; =\; 1\}^\{m\}\; |\; f\_\{i\}\; |\_\{p\_\{i.$

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. 251–260
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. 355–405 (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