Top Qs
Timeline
Chat
Perspective

Borell–Brascamp–Lieb inequality

Theorem in mathematics From Wikipedia, the free encyclopedia

Remove ads

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.

Remove ads

Statement of the inequality in Rn

Summarize
Perspective

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

where

and .

Then

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

Remove ads

References

  • Borell, Christer (1975). "Convex set functions in d-space". Period. Math. Hungar. 6 (2): 111–136. doi:10.1007/BF02018814.
  • Brascamp, Herm Jan & Lieb, Elliott H. (1976). "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 of Functional Analysis. 22 (4): 366–389. doi:10.1016/0022-1236(76)90004-5.
  • Cordero-Erausquin, Dario; McCann, Robert J. & Schmuckenschläger, Michael (2001). "A Riemannian interpolation inequality à la Borell, Brascamp and Lieb". Invent. Math. 146 (2): 219–257. doi:10.1007/s002220100160.
  • Gardner, Richard J. (2002). "The Brunn–Minkowski inequality" (PDF). Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.
  • Henstock, R.; Macbeath, A. M. (1953). "On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik". Proc. London Math. Soc. Series 3. 3: 182–194. doi:10.1112/plms/s3-3.1.182.
Remove ads
Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads