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Concept# Loomis–Whitney inequality

Summary

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^

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