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


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

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:


 * $$\left\|\prod_{j=1}^d g_j \circ \pi_j\right\|_{L^{1} (\mathbb{R}^{d })}

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


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

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

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. 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 $$\mathbb{R}^{d}$$ 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 $$j$$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:

$$|E|^{d-1}\leq 2^{-d}|\partial E|^d$$Iterating the theorem yields $$| E | \leq \prod_{1 \leq j < k \leq d} | \pi_{j}\circ \pi_k (E) |^{\binom{d-1}{2}^{-1}}$$ and more generally $$| E | \leq \prod_j | \pi_{j} (E) |^{\binom{d-1}{k}^{-1}}$$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 πj above are replaced by more general linear maps, not necessarily all mapping onto spaces of the same dimension.