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 inequality

Fix a dimension d ≥ 2 and consider the projections

$\pi_{j} : \mathbb{R}^{d} \to \mathbb{R}^{d - 1},$

$\pi_{j} : x = (x_{1}, \dots, x_{d}) \mapsto \hat{x}_{j} = (x_{1}, \dots, x_{j - 1}, x_{j + 1}, \dots, x_{d}).$

For each 1 ≤ j ≤ d, let

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

$g_{j} \in L^{d - 1} (\mathbb{R}^{d -1}).$

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

$f_{j} (x) = g_{j} (x)^{d - 1},$

$\int_{\mathbb{R}^{d}} \prod_{j = 1}^{d} f_{j} ( \pi_{j} (x) )^{1 / (d - 1)} \, \mathrm{d} x \leq \prod_{j = 1}^{d} \left( \int_{\mathbb{R}^{d - 1}} f_{j} (\hat{x}_{j}) \, \mathrm{d} \hat{x}_{j} \right)^{1 / (d - 1)}.$

A special case

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

$| E | \geq \prod_{j = 1}^{d} \frac{| E |}{| \pi_{j} (E) |}.$

The quantity

$\frac{| E |}{| \pi_{j} (E) |}$

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.

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, H. (1949). "An inequality related to the isoperimetric inequality". Bull. Amer. Math. Soc. 55: 961–962. doi:10.1090/S0002-9904-1949-09320-5. MR 0031538