Loomis-whitney Inequality

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The result is named after the American Mathematicians L. H. Loomis and Hassler Whitney , and was published in 1949. STATEMENT OF THE INEQUALITY Fix a dimension ''d'' ≥ 2 and consider the projections :$\backslash pi\_\{j\}\; :\; \backslash mathbb\{R\}^\{d\}\; o\; \backslash mathbb\{R\}^\{d\; -\; 1\},$ :$\backslash pi\_\{j\}\; :\; x\; =\; (x\_\{1\},\; \backslash dots,\; x\_\{d\})\; \backslash mapsto\; \backslash hat\{x\}\_\{j\}\; =\; (x\_\{1\},\; \backslash dots,\; x\_\{j\; -\; 1\},\; x\_\{j\; +\; 1\},\; \backslash dots,\; x\_\{d\}).$ For each 1 ≤ ''j'' ≤ ''d'', let :$g\_\{j\}\; :\; \backslash mathbb\{R\}^\{d\; -\; 1\}\; o\; [0,\; +\; \backslash infty),$ :$g\_\{j\}\; \backslash in\; L^\{d\; -\; 1\}\; (\backslash mathbb\{R\}^\{d\; -1\}).$ Then the Loomis-Whitney inequality holds:

A SPECIAL CASE


The Loomis-Whitney inequality can be used to relate the Lebesgue Measure of a subset of Euclidean Space $\backslash mathbb\{R\}^\{d\}$ to its "average widths" in the coordinate directions. Let ''E'' be some Measurable Subset of $\backslash mathbb\{R\}^\{d\}$ and let

:$f\_\{j\}\; =\; \backslash mathbf\{1\}\_\{\backslash pi\_\{j\}\; (E)\}$

be the Indicator Function of the projection of ''E'' onto the ''j''th coordinate hyperplane. It follows that for any point ''x'' in ''E'',

:$\backslash prod\_\{j\; =\; 1\}^\{d\}\; f\_\{j\}\; (\backslash pi\_\{j\}\; (x))^\{1\; /\; (d\; -\; 1)\}\; =\; 1.$

Hence, by the Loomis-Whitney inequality,