In mathematics, the Brascamp–Lieb inequality can refer to two inequalities. The first is a result in geometry concerning integrable functions on n-dimensional Euclidean space
Contents
The geometric inequality
Fix natural numbers m and n. For 1 ≤ i ≤ m, let ni ∈ N and let ci > 0 so that
Choose non-negative, integrable functions
and surjective linear maps
Then the following inequality holds:
where D is given by
Another way to state this is that the constant D is what one would obtain by restricting attention to the case in which each
The geometric Brascamp–Lieb inequality
The geometric Brascamp–Lieb inequality is a special case of the above, and was used by Ball (1989) to provide upper bounds for volumes of central sections of cubes.
For i = 1, ..., m, let ci > 0 and let ui ∈ Sn−1 be a unit vector; suppose that ci and ui satisfy
for all x in Rn. Let fi ∈ L1(R; [0, +∞]) for each i = 1, ..., m. Then
The geometric Brascamp–Lieb inequality follows from the Brascamp–Lieb inequality as stated above by taking ni = 1 and Bi(x) = x · ui. Then, for zi ∈ R,
It follows that D = 1 in this case.
Hölder's inequality
As another special case, take ni = n, Bi = id, the identity map on
i, and let ci = 1 / pi for 1 ≤ i ≤ m. Then
and the log-concavity of the determinant of a positive definite matrix implies that D = 1. This yields Hölder's inequality in
The concentration inequality
Consider a probability density function
Formally, let
where H is the Hessian and
Relationship with other inequalities
The Brascamp–Lieb inequality is an extension of the Poincaré inequality which only concerns Gaussian probability distributions.
The Brascamp–Lieb inequality is also related to the Cramér–Rao bound. While Brascamp–Lieb is an upper-bound, the Cramér–Rao bound lower-bounds the variance of