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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.

Statement of the inequality in R

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

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

where

M p ( a , b , λ ) = { ( ( 1 − λ ) a p + λ b p ) 1 / p if a b ≠ 0 0 if a b = 0 {\displaystyle {\begin{aligned}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{aligned}}}

and M 0 ( a , b , λ ) = a 1 − λ b λ {\displaystyle M_{0}(a,b,\lambda )=a^{1-\lambda }b^{\lambda }} .

Then

∫ R n h ( x ) d x ≥ M p / ( n p + 1 ) ( ∫ R n f ( x ) d x , ∫ R n g ( x ) d x , λ ) . {\displaystyle \int _{\mathbb {R} ^{n}}h(x)\,\mathrm {d} x\geq M_{p/(np+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.)

References

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