# Brascamp–Lieb inequality

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 $\mathbb {R} ^{n}$. 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 n$_i$ ∈ N and let c$_i$ > 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:
    $\int _{\mathbb {R} ^{n}}\prod _{i=1}^{m}f_{i}\left(B_{i}x\right)^{c_{i}}\,\mathrm {d} x\leq D^{-1/2}\prod _{i=1}^{m}\left(\int _{\mathbb {R} ^{n_{i}}}f_{i}(y)\,\mathrm {d} y\right)^{c_{i}},$
where D is given by
    $D=\inf \left\{\left.{\frac {\det \left(\sum _{i=1}^{m}c_{i}B_{i}^{*}A_{i}B_{i}\right)}{\prod _{i=1}^{m}(\det A_{i})^{c_{i}}}}\right|A_{i}{\text{ is a positive-definite }}n_{i}\times n_{i}{\text{ matrix}}\right\}.$
Another way to state this is that the constant D is what one would obtain by restricting attention to the case in which each $f_{i}$ is a centered Gaussian function, namely $f_{i}(y)=\exp\{-(y,\,A_{i}\,y)\}$.

## Alternative forms

Consider a probability density function $p(x)=\exp(-\phi (x))$. This probability density function $p(x)$ is said to be a log-concave measure if the $\phi (x)$ 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 $p(x)$. The Brascamp–Lieb inequality gives another characterization of the compactness of $p(x)$ by bounding the mean of any statistic $S(x)$.
Formally, let $S(x)$ be any derivable function. The Brascamp–Lieb inequality reads:
    $\operatorname {var} _{p}(S(x))\leq E_{p}(\nabla ^{T}S(x)[H\phi (x)]^{-1}\nabla S(x))$
where H is the Hessian and $\nabla$ 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)
* $d,n\geq 1$.
* $d_{1},...,d_{n}\in \{1,2,...,d\}$.
* $p_{1},...,p_{n}\in [0,\infty )$.
* $B_{i}:\mathbb {R} ^{d}\to \mathbb {R} ^{d_{i}}$ are linear surjections, with zero common kernel: $\cap _{i}ker(B_{i})=\{0\}$.
* Call $(B,p)=(B_{1},...,B_{n},p_{1},...,p_{n})$ a Brascamp-Lieb datum (BL datum).
For any $f_{i}\in L^{1}(R^{d_{i}})$ with $f_{i}\geq 0$, define$BL(B,p,f):={\frac {\int _{H}\prod _{j=1}^{m}\left(f_{j}\circ B_{j}\right)^{p_{j}}}{\prod _{j=1}^{m}\left(\int _{H_{j}}f_{j}\right)^{p_{j}}}}$
Now define the Brascamp-Lieb constant for the BL datum:$BL(B,p)=\max _{f}BL(B,p,f)$
Theorem — (BCCT, 2007)
$BL(B,p)$ is finite iff $d=\sum _{i}p_{i}d_{i}$, and for all subspace $V$ of $\mathbb {R} ^{d}$,
$dim(V)\leq \sum _{i}p_{i}dim(B_{i}(V))$
$BL(B,p)$ is reached by gaussians:
* If $BL(B,p)$ is finite, then there exists some linear operators $A_{i}:\mathbb {R} ^{d_{i}}\to \mathbb {R} ^{d_{i}}$ such that $f_{i}=e^{-\langle A_{i}x,x\rangle }$ achieves the upper bound.
* If $BL(B,p)$ is infinite, then there exists a sequence of gaussians for which
${\frac {\int _{H}\prod _{j=1}^{m}\left(f_{j}\circ B_{j}\right)^{p_{j}}}{\prod _{j=1}^{m}\left(\int _{H_{j}}f_{j}\right)^{p_{j}}}}\to \infty$

### Discrete case

Setup:
* Finitely generated abelian groups $G,G_{1},...,G_{n}$.
* Group homomorphisms $\phi _{j}:G\to G_{j}$.
* BL datum defined as $(G,G_{1},...,G_{n},\phi _{1},...\phi _{n})$
* $T(G)$ is the torsion subgroup, that is, the subgroup of finite-order elements.
With this setup, we have (Theorem 2.4, Theorem 3.12)
Theorem — If there exists some $s_{1},...,s_{n}\in [0,1]$ such that
$rank(H)\leq \sum _{j}s_{j}rank(\phi _{j}(H))\quad \forall H\leq G$
Then for all $0\geq f_{j}\in \ell ^{1/s_{j}}(G_{j})$,
$\left\|\prod _{j}f_{j}\circ \phi _{j}\right\|_{1}\leq |T(G)|\prod _{j}\|f_{j}\|_{1/s_{j}}$ and in particular,
$|E|\leq |T(G)|\prod _{j}|\phi _{j}(E)|^{s_{j}}\quad \forall E\subset G$
Note that the constant $|T(G)|$ is not always tight.

### BL polytope

Given BL datum $(B,p)$, the conditions for $BL(B,p)<\infty$ are
* $d=\sum _{i}p_{i}d_{i}$, and
* for all subspace $V$ of $\mathbb {R} ^{d}$,$dim(V)\leq \sum _{i}p_{i}dim(B_{i}(V))$
Thus, the subset of $p\in [0,\infty )^{n}$ 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 $V$ of $\mathbb {R} ^{d}$, there are only finitely many possible equations of $dim(V)\leq \sum _{i}p_{i}dim(B_{i}(V))$, 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 case of the Brascamp–Lieb inequality in which all the n$_i$ are equal to 1 was proved earlier than the general case. In 1989, Keith Ball introduced a "geometric form" of this inequality. Suppose that $(u_{i})_{i=1}^{m}$ are unit vectors in $\mathbb {R} ^{n}$ and $(c_{i})_{i=1}^{m}$ are positive numbers satisfying
    $\sum _{i=1}^{m}c_{i}\langle x,u_{i}\rangle u_{i}=x$
for all $x\in \mathbb {R} ^{n}$, and that $(f_{i})_{i=1}^{m}$ are positive measurable functions on $\mathbb {R}$. Then
    $\int _{\mathbb {R} ^{n}}\prod _{i=1}^{m}f_{i}(\langle x,u_{i}\rangle )^{c_{i}}\,\mathrm {d} x\leq \prod _{i=1}^{m}\left(\int _{\mathbb {R} }f_{i}(t)\,\mathrm {d} t\right)^{c_{i}}.$
Thus, when the vectors $(u_{i})$ resolve the inner product the inequality has a particularly simple form: the constant is equal to 1 and the extremal Gaussian densities are identical. Ball used this inequality to estimate volume ratios and isoperimetric quotients for convex sets in and.
There is also a geometric version of the more general inequality in which the maps $B_{i}$ are orthogonal projections and
    $\sum _{i=1}^{m}c_{i}B_{i}=I$
where $I$ is the identity operator on $\mathbb {R} ^{n}$.

### Hölder's inequality

Take n$_i$ = n, B$_i$ = id, the identity map on $\mathbb {R} ^{n}$, replacing f$_i$ by f$^{1/ci}$ $_i$, and let c$_i$ = 1 / p$_i$ 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 $\mathbb {R} ^{n}$:
    $\int _{\mathbb {R} ^{n}}\prod _{i=1}^{m}f_{i}(x)\,\mathrm {d} x\leq \prod _{i=1}^{m}\|f_{i}\|_{p_{i}}.$

### 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 $\operatorname {var} _{p}(S(x))$. The Cramér–Rao bound states
    $\operatorname {var} _{p}(S(x))\geq E_{p}(\nabla ^{T}S(x))[E_{p}(H\phi (x))]^{-1}E_{p}(\nabla S(x))\!$.
which is very similar to the Brascamp–Lieb inequality in the alternative form shown above.