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.
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 '''R'''<sup>''n''</sup>, :<math>h \left((1 - \lambda) x + \lambda y \right) \geq M_{p} \left(f(x), g(y), \lambda \right),</math> where :<math> \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>M_{0}(a,b,\lambda) = a^{1-\lambda}b^{\lambda}</math>. Then :<math>\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== * {{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 | doi = 10.1007/BF02018814 }} * {{cite journal |author1=Brascamp, Herm Jan |author2=Lieb, Elliott H. |name-list-style=amp | 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 = Journal of Functional Analysis | volume = 22 | year = 1976 | number = 4 | pages = 366–389 | doi = 10.1016/0022-1236(76)90004-5 |doi-access = free }} * {{cite journal | author = Cordero-Erausquin, Dario | author2 = McCann, Robert J. | author2-link = Robert McCann (mathematician) | author3 = Schmuckenschläger, Michael | name-list-style = amp | title = A Riemannian interpolation inequality à la Borell, Brascamp and Lieb | journal = Invent. Math. | volume = 146 | year = 2001 | number = 2 | pages = 219–257 | 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) | url = https://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 | doi-access=free }} * {{cite journal | author = Henstock, R. |author2=Macbeath, A. M. |author2-link=Alexander M. Macbeath | title = On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik | journal = Proc. London Math. Soc. |series=Series 3 | volume = 3 | year = 1953 | pages = 182–194 | doi = 10.1112/plms/s3-3.1.182 }} {{DEFAULTSORT:Borell-Brascamp-Lieb inequality}} [[Category:Geometric inequalities]]