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 $d\ge 2$ and consider the projections

:$\pi_ : \mathbb^ \to \mathbb^,$
:$\pi_ : x = (x_, \dots, x_) \mapsto \hat_ = (x_, \dots, x_, x_, \dots, x_).$

For each 1 ≤ ''j'' ≤ ''d'', let

:$g_ : \mathbb^ \to [0, + \infty),$
:$g_ \in L^ (\mathbb^).$

Then the Loomis–Whitney inequality holds:

:$\left\, \prod_^d g_j \circ \pi_j\right\, _ = \int_ \prod_^ g_ ( \pi_ (x) ) \, \mathrm x \leq \prod_^ \, g_ \, _.$

Equivalently, taking $f_ (x) = g_ (x)^,$ we have

:$f_ : \mathbb^ \to [0, + \infty),$
:$f_ \in L^ (\mathbb^)$

implying
:$\int_ \prod_^ f_ ( \pi_ (x) )^ \, \mathrm x \leq \prod_^ \left( \int_ f_ (\hat_) \, \mathrm \hat_ \right)^.$

A special case

The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space $\mathbb^$ 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 $\mathbb^$ and let

:$f_ = \mathbf_$

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

:$\prod_^ f_ (\pi_ (x))^ = \prod_^ 1 = 1.$

Hence, by the Loomis–Whitney inequality,

:$\int_ \mathbf 1_E(x) \, \mathrm x = , E , \leq \prod_^ , \pi_ (E) , ^,$

and hence

:$, E , \geq \prod_^ \frac.$

The quantity

:$\frac$

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

Corollary. Since $2 , \pi_j(E), \leq , \partial E,$, we get a loose isoperimetric inequality:

$, E, ^\leq 2^, \partial E, ^d$Iterating the theorem yields $, E , \leq \prod_ , \pi_\circ \pi_k (E) , ^$ and more generally$, E , \leq \prod_j , \pi_ (E) , ^$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.

References

Sources

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{{DEFAULTSORT:Loomis-Whitney inequality
Incidence geometry
Geometric inequalities