Brascamp–Lieb inequality

In mathematics, the Brascamp–Lieb inequality is a result in geometry concerning integrable functions on n-dimensional Euclidean space Rn. It generalizes the Loomis–Whitney inequality and Hölder's inequality, and is named after Herm Jan Brascamp and Elliott H. Lieb. The original inequality (called the geometric inequality here) is in .[1] Its generalization, stated first, is in [2]

Statement of the inequality

Fix natural numbers m and n. For 1 ≤ i ≤ m, let ni ∈ N and let ci > 0 so that

$\sum_{i = 1}^{m} c_{i} n_{i} = n.$

Choose non-negative, integrable functions

$f_{i} \in L^{1} \left( \mathbb{R}^{n_{i}} ; [0, + \infty] \right)$
$B_{i} : \mathbb{R}^{n} \to \mathbb{R}^{n_{i}}.$

Then the following inequality holds:

$\int_{\mathbb{R}^{n}} \prod_{i = 1}^{m} f_{i} \left( B_{i} x \right)^{c_{i}} \, \mathrm{d} x \leq D^{- 1/2} \prod_{i = 1}^{m} \left( \int_{\mathbb{R}^{n_{i}}} f_{i} (y) \, \mathrm{d} y \right)^{c_{i}},$

where D is given by

$D = \inf \left\{ \left. \frac{\det \left( \sum_{i = 1}^{m} c_{i} B_{i}^{*} A_{i} B_{i} \right)}{\prod_{i = 1}^{m} ( \det A_{i} )^{c_{i}}} \right| A_{i} \mbox{ is a positive-definite } n_{i} \times n_{i} \mbox{ matrix} \right\}.$

Another way to state this is that the constant D is what one would obtain by restricting attention to the case in which each $f_{i}$ is a centered Gaussian function, namely $f_{i}(y) = \exp \{-(y,\, A_{i}\, y)\}.$

Relationships to other inequalities

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 that ci and ui satisfy

$x = \sum_{i = 1}^{m} c_{i} (x \cdot u_{i}) u_{i}$

for all x in Rn. Let fi ∈ L1(R; [0, +∞]) for each i = 1, ..., m. Then

$\int_{\mathbb{R}^{n}} \prod_{i = 1}^{m} f_{i} (x \cdot u_{i})^{c_{i}} \, \mathrm{d} x \leq \prod_{i = 1}^{m} \left( \int_{\mathbb{R}} f_{i} (y) \, \mathrm{d} y \right)^{c_{i}}.$

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,

$B_{i}^{*} (z_{i}) = z_{i} u_{i}.$

It follows that D = 1 in this case.

Hölder's inequality

As another special case, take ni = n, Bi = id, the identity map on Rn, replacing fi by f1/ci
i
, and let ci = 1 / pi for 1 ≤ i ≤ m. Then

$\sum_{i = 1}^{m} \frac{1}{p_{i}} = 1$

and the log-concavity of the determinant of a positive definite matrix implies that D = 1. This yields Hölder's inequality in Rn:

$\int_{\mathbb{R}^{n}} \prod_{i = 1}^{m} f_{i} (x) \, \mathrm{d} x \leq \prod_{i = 1}^{m} \| f_{i} \|_{p_{i}}.$

References

1. ^ H.J. Brascamp and E.H. Lieb, Best Constants in Young's Inequality, Its Converse and Its Generalization to More Than Three Functions, Adv. in Math. 20, 151–172 (1976).
2. ^ E.H.Lieb, Gaussian Kernels have only Gaussian Maximizers, Inventiones Mathematicae 102, pp. 179–208 (1990).