Brascamp–Lieb inequality

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

${\displaystyle \sum _{i=1}^{m}c_{i}n_{i}=n.}$

Choose non-negative, integrable functions

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

and surjective linear maps

${\displaystyle B_{i}:\mathbb {R} ^{n}\to \mathbb {R} ^{n_{i}}.}$

Then the following inequality holds:

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

where D is given by

${\displaystyle D=\inf \left\{\left.{\frac {\det \left(\sum _{i=1}^{m}c_{i}B_{i}^{*}A_{i}B_{i}\right)}{\prod _{i=1}^{m}(\det A_{i})^{c_{i}}}}\right|A_{i}{\text{ is a positive-definite }}n_{i}\times n_{i}{\text{ matrix}}\right\}.}$

Another way to state this is that the constant D is what one would obtain by restricting attention to the case in which each ${\displaystyle f_{i}}$ is a centered Gaussian function, namely ${\displaystyle f_{i}(y)=\exp\{-(y,\,A_{i}\,y)\}}$.[1]

Consider a probability density function ${\displaystyle p(x)=\exp(-\phi (x))}$. This probability density function ${\displaystyle p(x)}$ is said to be a log-concave measure if the ${\displaystyle \phi (x)}$ 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 ${\displaystyle p(x)}$. The Brascamp–Lieb inequality gives another characterization of the compactness of ${\displaystyle p(x)}$ by bounding the mean of any statistic ${\displaystyle S(x)}$.

Formally, let ${\displaystyle S(x)}$ be any derivable function. The Brascamp–Lieb inequality reads:

${\displaystyle \operatorname {var} _{p}(S(x))\leq E_{p}(\nabla ^{T}S(x)[H\phi (x)]^{-1}\nabla S(x))}$

where H is the Hessian and ${\displaystyle \nabla }$ is the Nabla symbol.[2]

The inequality is generalized in 2008[3] 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)

• ${\displaystyle d,n\geq 1}$.

• ${\displaystyle d_{1},...,d_{n}\in \{1,2,...,d\}}$.

• ${\displaystyle p_{1},...,p_{n}\in [0,\infty )}$.

• ${\displaystyle B_{i}:\mathbb {R} ^{d}\to \mathbb {R} ^{d_{i}}}$ are linear surjections, with zero common kernel: ${\displaystyle \cap _{i}ker(B_{i})=\{0\}}$.
• Call ${\displaystyle (B,p)=(B_{1},...,B_{n},p_{1},...,p_{n})}$ a Brascamp-Lieb datum (BL datum).

For any ${\displaystyle f_{i}\in L^{1}(R^{d_{i}})}$ with ${\displaystyle f_{i}\geq 0}$, define

$${\displaystyle BL(B,p,f):={\frac {\int _{H}\prod _{j=1}^{m}\left(f_{j}\circ B_{j}\right)^{p_{j}}}{\prod _{j=1}^{m}\left(\int _{H_{j}}f_{j}\right)^{p_{j}}}}}$$

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

$${\displaystyle BL(B,p)=\max _{f}BL(B,p,f)}$$

Theorem — (BCCT, 2007)

${\displaystyle BL(B,p)}$ is finite iff ${\displaystyle d=\sum _{i}p_{i}d_{i}}$, and for all subspace ${\displaystyle V}$ of ${\displaystyle \mathbb {R} ^{d}}$,

$${\displaystyle dim(V)\leq \sum _{i}p_{i}dim(B_{i}(V))}$$

${\displaystyle BL(B,p)}$ is reached by gaussians:

• If ${\displaystyle BL(B,p)}$ is finite, then there exists some linear operators ${\displaystyle A_{i}:\mathbb {R} ^{d_{i}}\to \mathbb {R} ^{d_{i}}}$ such that ${\displaystyle f_{i}=e^{-\langle A_{i}x,x\rangle }}$ achieves the upper bound.

• If ${\displaystyle BL(B,p)}$ is infinite, then there exists a sequence of gaussians for which

$${\displaystyle {\frac {\int _{H}\prod _{j=1}^{m}\left(f_{j}\circ B_{j}\right)^{p_{j}}}{\prod _{j=1}^{m}\left(\int _{H_{j}}f_{j}\right)^{p_{j}}}}\to \infty }$$

Discrete case

Setup:

• Finitely generated abelian groups ${\displaystyle G,G_{1},...,G_{n}}$.

• Group homomorphisms ${\displaystyle \phi _{j}:G\to G_{j}}$.

• BL datum defined as ${\displaystyle (G,G_{1},...,G_{n},\phi _{1},...\phi _{n})}$

• ${\displaystyle T(G)}$ is the torsion subgroup, that is, the subgroup of finite-order elements.

With this setup, we have (Theorem 2.4,[4] Theorem 3.12 [5])

Theorem — If there exists some ${\displaystyle s_{1},...,s_{n}\in [0,1]}$ such that

$${\displaystyle rank(H)\leq \sum _{j}s_{j}rank(\phi _{j}(H))\quad \forall H\leq G}$$

Then for all ${\displaystyle 0\geq f_{j}\in \ell ^{1/s_{j}}(G_{j})}$,

$${\displaystyle \left\|\prod _{j}f_{j}\circ \phi _{j}\right\|_{1}\leq |T(G)|\prod _{j}\|f_{j}\|_{1/s_{j}}}$$

and in particular,

$${\displaystyle |E|\leq |T(G)|\prod _{j}|\phi _{j}(E)|^{s_{j}}\quad \forall E\subset G}$$

Note that the constant ${\displaystyle |T(G)|}$ is not always tight.

• Gardner, Richard J. (2002). "The Brunn–Minkowski inequality" (PDF). Bulletin of the American Mathematical Society. New Series. 39 (3): 355–405. doi:10.1090/S0273-0979-02-00941-2.