Loomis–Whitney inequality

Template:Short description 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 $d\ge 2$ and consider the projections

${\pi }_j:{ℝ}^d\to {ℝ}^{d-1},$

${\pi }_j:x=(x_1,\dots ,x_d)\mapsto {\hat{x}}_j=(x_1,\dots ,x_{j-1},x_{j+1},\dots ,x_d).$

For each 1 ≤ j ≤ d, let

$g_j:{ℝ}^{d-1}\to [0,+\mathrm{\infty }),$

$g_j\in L^{d-1}({ℝ}^{d-1}).$

Then the Loomis–Whitney inequality holds:

${\Vert \prod _{j=1}^d{g}_j\circ {\pi }_j\Vert }_{L^1({ℝ}^d)}=\underset{{ℝ}^d}{\int }\prod _{j=1}^d{g}_j({\pi }_j(x))\mathrm{d}x\le \prod _{j=1}^d\Vert g_j{\Vert }_{L^{d-1}({ℝ}^{d-1})}.$

Equivalently, taking $f_j(x)=g_j(x{)}^{d-1},$ we have

$f_j:{ℝ}^{d-1}\to [0,+\mathrm{\infty }),$

$f_j\in L^1({ℝ}^{d-1})$

implying

$\underset{{ℝ}^d}{\int }\prod _{j=1}^d{f}_j({\pi }_j(x){)}^{1/(d-1)}\mathrm{d}x\le \prod _{j=1}^d{(\underset{{ℝ}^{d-1}}{\int }{f}_j({\hat{x}}_j)\mathrm{d}{\hat{x}}_j)}^{1/(d-1)}.$

A special case

The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space ${ℝ}^d$ 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 ${ℝ}^d$ and let

$f_j={\mathrm{𝟏}}_{{\pi }_j(E)}$

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,

$\underset{{ℝ}^d}{\int }{\mathrm{𝟏}}_E(x)\mathrm{d}x=|E|\le \prod _{j=1}^d|{\pi }_j(E){|}^{1/(d-1)},$

and hence

$|E|\ge \prod _{j=1}^d\frac{|E|}{|{\pi }_j(E)|}.$

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

Corollary. Since $2|{\pi }_j(E)|\le |\partial E|$, we get a loose isoperimetric inequality:

$$|E{|}^{d-1}\le 2^{-d}|\partial E{|}^d$$Iterating the theorem yields $|E|\le \prod _{1\le j<k\le d}|{\pi }_j\circ {\pi }_k(E){|}^{{(\genfrac{}{}{0ex}{}{d-1}{2})}^{-1}}$ and more generally$$|E|\le \prod _j|{\pi }_j(E){|}^{{(\genfrac{}{}{0ex}{}{d-1}{k})}^{-1}}$$where ${\pi }_j$ enumerates over all projections of ${ℝ}^d$ 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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