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

For each 1 ≤ jd, let

Then the Loomis–Whitney inequality holds:

Equivalently, taking

A special case[edit]

The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space to its "average widths" in the coordinate directions. Let E be some measurable subset of and let

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

Hence, by the Loomis–Whitney inequality,

and hence

The quantity

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]