# Loomis–Whitney inequality

In mathematics, the Loomis–Whitney inequality is a result in geometry, which in its simplest form, allows one to estimate the "size" of a $d$-dimensional set by the sizes of its $(d-1)$-dimensional projections. The inequality has applications in incidence geometry, the study of so-called "lattice animals", and other areas.
The result is named after the American mathematicians Lynn Harold Loomis and Hassler Whitney, and was published in 1949.

## Statement of the inequality

Fix a dimension $d\geq 2$ and consider the projections
    $\pi _{j}:\mathbb {R} ^{d}\to \mathbb {R} ^{d-1},$
    $\pi _{j}:x=(x_{1},\dots ,x_{d})\mapsto {\hat {x}}_{j}=(x_{1},\dots ,x_{j-1},x_{j+1},\dots ,x_{d}).$
For each 1 ≤ j ≤ d, let
    $g_{j}:\mathbb {R} ^{d-1}\to [0,+\infty ),$
    $g_{j}\in L^{d-1}(\mathbb {R} ^{d-1}).$
Then the Loomis–Whitney inequality holds:
    $\left\|\prod _{j=1}^{d}g_{j}\circ \pi _{j}\right\|_{L^{1}(\mathbb {R} ^{d})}=\int _{\mathbb {R} ^{d}}\prod _{j=1}^{d}g_{j}(\pi _{j}(x))\,\mathrm {d} x\leq \prod _{j=1}^{d}\|g_{j}\|_{L^{d-1}(\mathbb {R} ^{d-1})}.$
Equivalently, taking $f_{j}(x)=g_{j}(x)^{d-1},$ we have
    $f_{j}:\mathbb {R} ^{d-1}\to [0,+\infty ),$
    $f_{j}\in L^{1}(\mathbb {R} ^{d-1})$
implying
    $\int _{\mathbb {R} ^{d}}\prod _{j=1}^{d}f_{j}(\pi _{j}(x))^{1/(d-1)}\,\mathrm {d} x\leq \prod _{j=1}^{d}\left(\int _{\mathbb {R} ^{d-1}}f_{j}({\hat {x}}_{j})\,\mathrm {d} {\hat {x}}_{j}\right)^{1/(d-1)}.$

## A special case

The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space $\mathbb {R} ^{d}$ to its "average widths" in the coordinate directions. This is in fact the original version published by Loomis and Whitney in 1949 (the above is a generalization).
Let E be some measurable subset of $\mathbb {R} ^{d}$ and let
    $f_{j}=\mathbf {1} _{\pi _{j}(E)}$
be the indicator function of the projection of E onto the jth coordinate hyperplane. It follows that for any point x in E,
    $\prod _{j=1}^{d}f_{j}(\pi _{j}(x))^{1/(d-1)}=\prod _{j=1}^{d}1=1.$
Hence, by the Loomis–Whitney inequality,
    $\int _{\mathbb {R} ^{d}}\mathbf {1} _{E}(x)\,\mathrm {d} x=|E|\leq \prod _{j=1}^{d}|\pi _{j}(E)|^{1/(d-1)},$
and hence
    $|E|\geq \prod _{j=1}^{d}{\frac {|E|}{|\pi _{j}(E)|}}.$
The quantity
    ${\frac {|E|}{|\pi _{j}(E)|}}$
can be thought of as the average width of $E$ in the $j$th coordinate direction. This interpretation of the Loomis–Whitney inequality also holds if we consider a finite subset of Euclidean space and replace Lebesgue measure by counting measure.
The following proof is the original one
Overview: We prove it for unions of unit cubes on the integer grid, then take the continuum limit. When $d=1,2$, it is obvious. Now induct on $d+1$. The only trick is to use Hölder's inequality for counting measures.
Enumerate the dimensions of $\mathbb {R} ^{d+1}$ as $0,1,...,d$.
Given $N$ unit cubes on the integer grid in $\mathbb {R} ^{d+1}$, with their union being $T$, we project them to the 0-th coordinate. Each unit cube projects to an integer unit interval on $\mathbb {R}$. Now define the following:
* $I_{1},...,I_{k}$ enumerate all such integer unit intervals on the 0-th coordinate.
* Let $T_{i}$ be the set of all unit cubes that projects to $I_{i}$.
* Let $N_{j}$ be the area of $\pi _{j}(T)$, with $j=0,1,...,d$.
* Let $a_{i}$ be the volume of $T_{i}$. We have $\sum _{i}a_{i}=N$, and $a_{i}\leq N_{0}$.
* Let $T_{ij}$ be $\pi _{j}(T_{i})$ for all $j=1,...,d$.
* Let $a_{ij}$ be the area of $T_{ij}$. We have $\sum _{i}a_{ij}=N_{j}$.
By induction on each slice of $T_{i}$, we have $a_{i}^{d-1}\leq \prod _{j=1}^{d}a_{ij}$
Multiplying by $a_{i}\leq N_{0}$, we have $a_{i}^{d}\leq N_{0}\prod _{j=1}^{d}a_{ij}$
Thus $N=\sum _{i}a_{i}\leq \sum _{i}N_{0}^{1/d}\prod _{j=1}^{d}a_{ij}^{1/d}=N_{0}^{1/d}\sum _{i=1}^{k}\prod _{j=1}^{d}a_{ij}^{1/d}$
Now, the sum-product can be written as an integral over counting measure, allowing us to perform Holder's inequality: $\sum _{i=1}^{k}\prod _{j=1}^{d}a_{ij}^{1/d}=\int _{i}\prod _{j=1}^{d}a_{ij}^{1/d}=\left\|\prod _{j=1}^{d}a_{\cdot ,j}^{1/d}\right\|_{1}\leq \prod _{j}\|a_{\cdot ,j}^{1/d}\|_{d}=\prod _{j=1}^{d}\left(\sum _{i=1}^{k}a_{ij}\right)^{1/d}$
Plugging in $\sum _{i}a_{ij}=N_{j}$, we get $N^{d}\leq \prod _{j=0}^{d}N_{j}$
Corollary. Since $2|\pi _{j}(E)|\leq |\partial E|$, we get a loose isoperimetric inequality:
$|E|^{d-1}\leq 2^{-d}|\partial E|^{d}$Iterating the theorem yields $|E|\leq \prod _{1\leq j<k\leq d}|\pi _{j}\circ \pi _{k}(E)|^{{\binom {d-1}{2}}^{-1}}$ and more generally$|E|\leq \prod _{j}|\pi _{j}(E)|^{{\binom {d-1}{k}}^{-1}}$where $\pi _{j}$ enumerates over all projections of $\mathbb {R} ^{d}$ to its $d-k$ dimensional subspaces.

## Generalizations

The Loomis–Whitney inequality is a special case of the Brascamp–Lieb inequality, in which the projections π$_j$ above are replaced by more general linear maps, not necessarily all mapping onto spaces of the same dimension.