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

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).^[1]

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)}=1.

Hence, by the Loomis–Whitney inequality,

|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^[1]

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^[2]

|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.

References

^
ISSN 0273-0979
.

ISBN 978-3-662-07441-1
.

Sources

Zbl 1333.05001
.

Boucheron, Stéphane; Lugosi, Gábor;
Zbl 1279.60005
.

Zbl 0633.53002
.

Zbl 0078.35703
.

Zbl 0035.38302
.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Loomis–Whitney_inequality&oldid=1184174976"

[:0-1] 
ISSN 0273-0979
.

[2] ISBN 978-3-662-07441-1
.

[1]

[2]