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

Proof

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.

Sources

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• Burago, Yu. D.; Zalgaller, V. A. (1988). Geometric inequalities. Grundlehren der mathematischen Wissenschaften. Vol. 285. Translated by Sosinskiĭ, A. B. Berlin: Springer-Verlag. doi:10.1007/978-3-662-07441-1. ISBN 3-540-13615-0. MR 0936419. Zbl 0633.53002.
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• Loomis, L. H.; Whitney, H. (1949). "An inequality related to the isoperimetric inequality". Bulletin of the American Mathematical Society. 55 (10): 961–962. doi:10.1090/S0002-9904-1949-09320-5. MR 0031538. Zbl 0035.38302.