Loomis–Whitney inequality

Fix a dimension $d\backslash ge\; 2$ and consider the projections

:$\backslash pi\_\{j\}\; :\; \backslash mathbb\{R\}^\{d\}\; \backslash to\; \backslash mathbb\{R\}^\{d\; -\; 1\},$

:$\backslash pi\_\{j\}\; :\; x\; =\; (x\_\{1\},\; \backslash dots,\; x\_\{d\})\; \backslash mapsto\; \backslash hat\{x\}\_\{j\}\; =\; (x\_\{1\},\; \backslash dots,\; x\_\{j\; -\; 1\},\; x\_\{j\; +\; 1\},\; \backslash dots,\; x\_\{d\}).$

For each 1 ≤ j ≤ d, let

:$g\_\{j\}\; :\; \backslash mathbb\{R\}^\{d\; -\; 1\}\; \backslash to\; [0,\; +\; \backslash infty),$

:$g\_\{j\}\; \backslash in\; L^\{d\; -\; 1\}\; (\backslash mathbb\{R\}^\{d\; -1\}).$

Then the Loomis–Whitney inequality holds:

:$\backslash left\backslash |\backslash prod\_\{j=1\}^d\; g\_j\; \backslash circ\; \backslash pi\_j\backslash right\backslash |\_\{L^\{1\}\; (\backslash mathbb\{R\}^\{d\; \})\}$

= \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

:$f\_\{j\}\; :\; \backslash mathbb\{R\}^\{d\; -\; 1\}\; \backslash to\; [0,\; +\; \backslash infty),$

:$f\_\{j\}\; \backslash in\; L^\{1\}\; (\backslash mathbb\{R\}^\{d\; -1\})$

implying

:$\backslash int\_\{\backslash mathbb\{R\}^\{d\}\}\; \backslash prod\_\{j\; =\; 1\}^\{d\}\; f\_\{j\}\; (\; \backslash pi\_\{j\}\; (x)\; )^\{1\; /\; (d\; -\; 1)\}\; \backslash ,\; \backslash mathrm\{d\}\; x\; \backslash leq\; \backslash prod\_\{j\; =\; 1\}^\{d\}\; \backslash left(\; \backslash int\_\{\backslash mathbb\{R\}^\{d\; -\; 1\}\}\; f\_\{j\}\; (\backslash hat\{x\}\_\{j\})\; \backslash ,\; \backslash mathrm\{d\}\; \backslash hat\{x\}\_\{j\}\; \backslash right)^\{1\; /\; (d\; -\; 1)\}.$

The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space $\backslash mathbb\{R\}^\{d\}$ 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).{{Cite journal |last1=Loomis |first1=L. H. |last2=Whitney |first2=H. |date=1949 |title=An inequality related to the isoperimetric inequality |url=https://www.ams.org/bull/1949-55-10/S0002-9904-1949-09320-5/ |journal=Bulletin of the American Mathematical Society |language=en |volume=55 |issue=10 |pages=961–962 |doi=10.1090/S0002-9904-1949-09320-5 |issn=0273-0979|doi-access=free }}

Let E be some measurable subset of $\backslash mathbb\{R\}^\{d\}$ and let

:$f\_\{j\}\; =\; \backslash mathbf\{1\}\_\{\backslash pi\_\{j\}\; (E)\}$

be the indicator function of the projection of E onto the jth coordinate hyperplane. It follows that for any point x in E,

:$\backslash prod\_\{j\; =\; 1\}^\{d\}\; f\_\{j\}\; (\backslash pi\_\{j\}\; (x))^\{1\; /\; (d\; -\; 1)\}\; =\; \backslash prod\_\{j\; =\; 1\}^\{d\}\; 1\; =\; 1.$

Hence, by the Loomis–Whitney inequality,

:$\backslash int\_\{\backslash mathbb\{R\}^\{d\}\}\; \backslash mathbf\; 1\_E(x)\; \backslash ,\; \backslash mathrm\{d\}\; x\; =\; |\; E\; |\; \backslash leq\; \backslash prod\_\{j\; =\; 1\}^\{d\}\; |\; \backslash pi\_\{j\}\; (E)\; |^\{1\; /\; (d\; -\; 1)\},$

and hence

:$|\; E\; |\; \backslash geq\; \backslash prod\_\{j\; =\; 1\}^\{d\}\; \backslash frac$

The quantity

:$\backslash frac$

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

{{Math proof|title=Proof|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 $\backslash R^\{d+1\}$ as $0,\; 1,\; ...,\; d$.

Given $N$ unit cubes on the integer grid in $\backslash R^\{d+1\}$, with their union being $T$, we project them to the 0-th coordinate. Each unit cube projects to an integer unit interval on $\backslash R$. 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 $\backslash pi\_j(T)$, with $j\; =\; 0,\; 1,\; ...,\; d$.

• Let $a\_i$ be the volume of $T\_i$. We have $\backslash sum\_i\; a\_i\; =\; N$, and $a\_i\; \backslash leq\; N\_0$.

• Let $T\_\{ij\}$ be $\backslash pi\_j(T\_i)$ for all $j\; =\; 1,\; ...,\; d$.

• Let $a\_\{ij\}$ be the area of $T\_\{ij\}$. We have $\backslash sum\_i\; a\_\{ij\}\; =\; N\_j$.

By induction on each slice of $T\_i$, we have $a\_i^\{d-1\}\backslash leq\; \backslash prod\_\{j=1\}^d\; a\_\{ij\}$

Multiplying by $a\_i\; \backslash leq\; N\_0$, we have $a\_i^\{d\}\backslash leq\; N\_0\backslash prod\_\{j=1\}^d\; a\_\{ij\}$

Thus

$$N\; =\; \backslash sum\_i\; a\_i\; \backslash leq\; \backslash sum\_i\; N\_0^\{1/d\}\; \backslash prod\_\{j=1\}^d\; a\_\{ij\}^\{1/d\}\; =\; N\_0^\{1/d\}\; \backslash sum\_\{i=1\}^k\backslash prod\_\{j=1\}^d\; a\_\{ij\}^\{1/d\}$$

Now, the sum-product can be written as an integral over counting measure, allowing us to perform Holder's inequality:

$$\backslash sum\_\{i=1\}^k\backslash prod\_\{j=1\}^d\; a\_\{ij\}^\{1/d\}\; =\; \backslash int\_i\; \backslash prod\_\{j=1\}^d\; a\_\{ij\}^\{1/d\}\; =\; \backslash left\backslash |\backslash prod\_\{j=1\}^d\; a\_\{\backslash cdot,\; j\}^\{1/d\}\backslash right\backslash |\_1\; \backslash leq\; \backslash prod\_j\; \backslash |a\_\{\backslash cdot,\; j\}^\{1/d\}\backslash |\_d=\backslash prod\_\{j=1\}^d\; \backslash left(\backslash sum\_\{i=1\}^k\; a\_\{ij\}\backslash right)^\{1/d\}$$

Plugging in $\backslash sum\_i\; a\_\{ij\}\; =\; N\_j$, we get $N^d\; \backslash leq\; \backslash prod\_\{j=0\}^d\; N\_j$

}}

Corollary. Since $2\; |\backslash pi\_j(E)|\; \backslash leq\; |\backslash partial\; E|$, we get a loose isoperimetric inequality:

$$|E|^\{d-1\}\backslash leq\; 2^\{-d\}|\backslash partial\; E|^d$$Iterating the theorem yields and more generally{{Cite book |last1=Burago |first1=Yurii D. |url=https://books.google.com/books?id=Gpz6CAAAQBAJ&pg=PA1 |title=Geometric Inequalities |last2=Zalgaller |first2=Viktor A. |date=2013-03-14 |publisher=Springer Science & Business Media |isbn=978-3-662-07441-1 |pages=95 |language=en}}$$|\; E\; |\; \backslash leq\; \backslash prod\_j\; |\; \backslash pi\_\{j\}\; (E)\; |^\{\backslash binom\{d-1\}\{k\}^\{-1\}\}$$where $\backslash pi\_j$ enumerates over all projections of $\backslash R^d$ to its $d-k$ dimensional subspaces.

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.

{{reflist}}

• {{cite book|mr=3524748|last1=Alon|first1=Noga|last2=Spencer|first2=Joel H.|title=The probabilistic method|edition=Fourth edition of 1992 original|series=Wiley Series in Discrete Mathematics and Optimization|publisher=John Wiley & Sons, Inc.|location=Hoboken, NJ|year=2016|isbn=978-1-119-06195-3|author-link1=Noga Alon|author-link2=Joel H. Spencer|zbl=1333.05001}}
• {{cite book|mr=3185193|last1=Boucheron|first1=Stéphane|last2=Lugosi|first2=Gábor|last3=Massart|first3=Pascal|author-link3=Pascal Massart|title=Concentration inequalities. A nonasymptotic theory of independence|publisher=Oxford University Press|location=Oxford|year=2013|isbn=978-0-19-953525-5|zbl=1279.60005|doi=10.1093/acprof:oso/9780199535255.001.0001}}
• {{cite book|mr=0936419|last1=Burago|first1=Yu. D.|last2=Zalgaller|first2=V. A.|title=Geometric inequalities|translator-first1=A. B.|translator-last1=Sosinskiĭ|series=Grundlehren der mathematischen Wissenschaften|volume=285|publisher=Springer-Verlag|location=Berlin|year=1988|isbn=3-540-13615-0|author-link1=Yuri Burago|author-link2=Victor Zalgaller|doi=10.1007/978-3-662-07441-1|zbl=0633.53002}}
• {{cite book|mr=0102775|last1=Hadwiger|first1=H.|title=Vorlesungen über Inhalt, Oberfläche und Isoperimetrie|publisher=Springer-Verlag|location=Berlin–Göttingen–Heidelberg|year=1957|zbl=0078.35703|isbn=3-642-94702-6|series=Grundlehren der mathematischen Wissenschaften|volume=93|doi=10.1007/978-3-642-94702-5|author-link1=Hugo Hadwiger}}
• {{cite journal

| last1 = Loomis

| first1 = L. H.

|authorlink1=Lynn Harold Loomis

|last2=Whitney

|first2=H.

|authorlink2=Hassler Whitney

| title = An inequality related to the isoperimetric inequality

| journal = Bulletin of the American Mathematical Society

| volume = 55

| issue = 10

| year = 1949

| pages = 961–962

| doi = 10.1090/S0002-9904-1949-09320-5

| doi-access = free

|mr=0031538

|zbl=0035.38302

}}

{{DEFAULTSORT:Loomis-Whitney inequality}}

Category:Incidence geometry

Category:Geometric inequalities