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 L. H. Loomis and Hassler Whitney, and was published in 1949.

tatement of the inequality

Fix a dimension "d" ≥ 2 and consider the projections

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

For each 1 ≤ "j" ≤ "d", let

:$g\_\{j\}\; :\; mathbb\{R\}^\{d\; -\; 1\}\; o\; [0,\; +\; infty),$:$g\_\{j\}\; in\; L^\{d\; -\; 1\}\; (mathbb\{R\}^\{d\; -1\}).$

Then the Loomis-Whitney inequality holds:

:$int\_\{mathbb\{R\}^\{d\; prod\_\{j\; =\; 1\}^\{d\}\; g\_\{j\}\; (\; pi\_\{j\}\; (x)\; )\; ,\; mathrm\{d\}\; x\; leq\; prod\_\{j\; =\; 1\}^\{d\}\; |\; g\_\{j\}\; |\_\{L^\{d\; -\; 1\}\; (mathbb\{R\}^\{d\; -\; 1\})\}.$

Equivalently, taking

:$f\_\{j\}\; (x)\; =\; g\_\{j\}\; (x)^\{d\; -\; 1\},$

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

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. 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 "j"th 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

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

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.

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

* cite journal
last = Loomis
first = Lynn H.
coauthors = Whitney, H.
title = An inequality related to the isoperimetric inequality
journal = Bull. Amer. Math. Soc.
volume = 55
year = 1949
pages = 961–962
doi = 10.1090/S0002-9904-1949-09320-5 MathSciNet|id=0031538