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 ${\displaystyle d}$-dimensional set by the sizes of its ${\displaystyle (d-1)}$-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 ${\displaystyle d\geq 2}$ 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 ≤ j ≤ d, 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 case

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 ${\displaystyle E}$ in the ${\displaystyle j}$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

• Loomis, Lynn H.; Whitney, Hassler (1949). "An inequality related to the isoperimetric inequality". Bulletin of the American Mathematical Society. 55 (10): 961–962. doi:10.1090/S0002-9904-1949-09320-5. MR0031538