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

:

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