Loomis–Whitney inequality

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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[edit]

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 ≤ jd, 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[edit]

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[edit]

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.

References[edit]