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.

Statement of the inequality in Rⁿ

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,

[math]\displaystyle{ h \left( (1 - \lambda) x + \lambda y \right) \geq M_{p} \left( f(x), g(y), \lambda \right), }[/math]

where

[math]\displaystyle{ \begin{align} M_{p} (a, b, \lambda) = \begin{cases} &\left( (1 - \lambda) a^{p} + \lambda b^{p} \right)^{1/p} \; \quad \text{if} \quad ab\neq 0\\ &0 \quad \text{if} \quad ab=0 \end{cases} \end{align} }[/math]

and [math]\displaystyle{ M_{0}(a,b,\lambda) = a^{1-\lambda}b^{\lambda} }[/math].

Then

[math]\displaystyle{ \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 \right). }[/math]

(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

• 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". Bull. Amer. Math. Soc. (N.S.) 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2. https://www.ams.org/bull/2002-39-03/S0273-0979-02-00941-2/S0273-0979-02-00941-2.pdf. 
• "On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik". Proc. London Math. Soc.. Series 3 3: 182–194. 1953. doi:10.1112/plms/s3-3.1.182.