In mathematics, the Brascamp–Lieb inequality is either of two inequalities. The first is a result in geometry concerning integrable functions on n-dimensional Euclidean space . It generalizes the Loomis–Whitney inequality and Hölder's inequality. The second is a result of probability theory which gives a concentration inequality for log-concave probability distributions. Both are named after Herm Jan Brascamp and Elliott H. Lieb.

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  is a centered Gaussian function, namely .

Alternative forms 
Consider a probability density function . This probability density function  is said to be a log-concave measure if the  function is convex. Such probability density functions have tails which decay exponentially fast, so most of the probability mass resides in a small region around the mode of . The Brascamp–Lieb inequality gives another characterization of the compactness of  by bounding the mean of any statistic .

Formally, let  be any derivable function. The Brascamp–Lieb inequality reads:

where H is the Hessian and  is the Nabla symbol.

BCCT inequality 
The inequality is generalized in 2008 to account for both continuous and discrete cases, and for all linear maps, with precise estimates on the constant.

Definition: the Brascamp-Lieb datum (BL datum)

 .

 .

 .

  are linear surjections, with zero common kernel: .
 Call  a Brascamp-Lieb datum (BL datum).

For any  with , define

Now define the Brascamp-Lieb constant for the BL datum:

Discrete case 

Setup: 

 Finitely generated abelian groups .

 Group homomorphisms .

 BL datum defined as 

  is the torsion subgroup, that is, the subgroup of finite-order elements.

With this setup, we have (Theorem 2.4, Theorem 3.12 )

Note that the constant  is not always tight.

BL polytope 
Given BL datum , the conditions for  are 

 , and 
 for all subspace  of ,

Thus, the subset of  that satisfies the above two conditions is a closed convex polytope defined by linear inequalities. This is the BL polytope.

Note that while there are infinitely many possible choices of subspace  of , there are only finitely many possible equations of , so the subset is a closed convex polytope.

Similarly we can define the BL polytope for the discrete case.

Relationships to other inequalities

The geometric Brascamp–Lieb inequality 
The geometric Brascamp–Lieb inequality, first derived in 1976, is a special case of the general inequality. It was used by Keith Ball, in 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 
Take ni = n, Bi = id, the identity map on , replacing fi by f, 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 :

Poincaré inequality 
The Brascamp–Lieb inequality is an extension of the Poincaré inequality which only concerns Gaussian probability distributions.

Cramér–Rao bound 
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 . The Cramér–Rao bound states

.
which is very similar to the Brascamp–Lieb inequality in the alternative form shown above.

References

 

Geometric inequalities