Borell-Brascamp-Lieb inequality

Borell-Brascamp-Lieb inequality

In mathematics, the Borell-Brascamp-Lieb inequality is an integral inequality due to many different mathematicians but named after Christer Borell, Herm Jan Brascamp and Elliott Lieb.

The result was proved for "p" > 0 by Henstock and Macbeath in 1953. The case "p" = 0 is known as the Prékopa-Leindler inequality and was re-discovered by Brascamp and Lieb in 1976, when they proved the general version below; working independently, Borell had done the same in 1975. The nomenclature of "Borell-Brascamp-Lieb inequality" is due to Cordero-Erausquin, McCann and Schmuckenschläger, who in 2001 generalized the result to Riemannian manifolds such as the sphere and hyperbolic space.

tatement of the inequality in R"n"

Let 0 < "λ" < 1, let −1 / "n" ≤ "p" ≤ +∞, and let "f", "g", "h" : R"n" → [0, +∞) be integrable functions such that, for all "x" and "y" in R"n",

:h left( (1 - lambda) x + lambda y ight) geq M_{p} left( f(x), g(y), lambda ight),

where

:egin{align}M_{p} (a, b, lambda)& = left( (1 - lambda) a^{p} + lambda b^{p} ight)^{1/p},\M_{0} (a, b, lambda)& = a^{1 - lambda} b^{lambda}.,end{align}

Then

:int_{mathbb{R}^{n h(x) , mathrm{d} x geq M_{p / (n p + 1)} left( int_{mathbb{R}^{n f(x) , mathrm{d} x, int_{mathbb{R}^{n g(x) , mathrm{d} x, lambda ight).

(When "p" = −1 / "n", the convention is to take "p" / ("n" "p" + 1) to be −∞; when "p" = +∞, it is taken to be 1 / "n".)

References

* cite journal
last = Borell
first = Christer
title = Convex set functions in "d"-space
journal = Period. Math. Hungar.
volume = 6
year = 1975
number = 2
pages = 111–136
issn = 0031-5303
doi = 10.1007/BF02018814

* cite journal
author = Brascamp, Herm Jan and Lieb, Elliott H.
title = On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation
journal = J. Functional Analysis
volume = 22
year = 1976
number = 4
pages = 366–389
doi = 10.1016/0022-1236(76)90004-5

* cite journal
author = Cordero-Erausquin, Dario, McCann, Robert J. and Schmuckenschläger, Michael
title = A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
journal = Invent. Math.
volume = 146
year = 2001
number = 2
pages = 219–257
issn = 0020-9910
doi = 10.1007/s002220100160

* 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=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

* cite journal
author = Henstock, R. and Macbeath, A. M.
title = On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik
journal = Proc. London Math. Soc. (3)
volume = 3
year = 1953
pages = 182–194
issn = 0024-6115
doi = 10.1112/plms/s3-3.1.182


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