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


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