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.

Fix a dimension d ≥ 2 and consider the projections

For each 1 ≤ j ≤ d, let

Then the Loomis–Whitney inequality holds:

Equivalently, taking

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

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