Brascamp-Lieb inequality

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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, the Prékopa-Leindler inequality and Hölder's inequality, and is named after Herm Jan Brascamp and Elliott Lieb.

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[edit] 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)

and surjective linear maps

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} (x) \, \mathrm{d} x \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\}.

[edit] Relationships to other inequalities

[edit] 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}^{n}} f_{i} (x) \, \mathrm{d} x \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.

[edit] Hölder's inequality

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

\sum_{i = 1}^{m} 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}}.

[edit] The Prékopa-Leindler inequality

The Brascamp-Lieb inequality implies the Prékopa-Leindler inequality as the special case m = 2, n1 = n2 = n, B1 = B2 = id, c1 = (1 − λ) and c2 = λ.

[edit] References