Borell-Brascamp-Lieb inequality

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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.

[edit] 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,

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

where

\begin{align} M_{p} (a, b, \lambda)& = \left( (1 - \lambda) a^{p} + \lambda b^{p} \right)^{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 \right).

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

[edit] References