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.

Statement of the inequalityEdit

Fix a dimension d ≥ 2 and consider the projections

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

For each 1 ≤ jd, let

${\displaystyle g_{j}:\mathbb {R} ^{d-1}\to [0,+\infty ),}$
${\displaystyle g_{j}\in L^{d-1}(\mathbb {R} ^{d-1}).}$

Then the Loomis–Whitney inequality holds:

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

Equivalently, taking

${\displaystyle f_{j}(x)=g_{j}(x)^{d-1},}$
${\displaystyle \int _{\mathbb {R} ^{d}}\prod _{j=1}^{d}f_{j}(\pi _{j}(x))^{1/(d-1)}\,\mathrm {d} x\leq \prod _{j=1}^{d}\left(\int _{\mathbb {R} ^{d-1}}f_{j}({\hat {x}}_{j})\,\mathrm {d} {\hat {x}}_{j}\right)^{1/(d-1)}.}$

A special caseEdit

The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space ${\displaystyle \mathbb {R} ^{d}}$  to its "average widths" in the coordinate directions. Let E be some measurable subset of ${\displaystyle \mathbb {R} ^{d}}$  and let

${\displaystyle f_{j}=\mathbf {1} _{\pi _{j}(E)}}$

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

${\displaystyle \prod _{j=1}^{d}f_{j}(\pi _{j}(x))^{1/(d-1)}=1.}$

Hence, by the Loomis–Whitney inequality,

${\displaystyle |E|\leq \prod _{j=1}^{d}|\pi _{j}(E)|^{1/(d-1)},}$

and hence

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

The quantity

${\displaystyle {\frac {|E|}{|\pi _{j}(E)|}}}$

can be thought of as the average width of E in the jth 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.

GeneralizationsEdit

The Loomis–Whitney inequality is a special case of the Brascamp–Lieb inequality, in which the projections πj above are replaced by more general linear maps, not necessarily all mapping onto spaces of the same dimension.