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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Loomis–Whitney inequality ： ウィキペディア英語版 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. ==Statement of the inequality== Fix a dimension ''d'' ≥ 2 and consider the projections :$\backslash pi\_\; :\; \backslash mathbb^\; \backslash to\; \backslash mathbb^,$ :$\backslash pi\_\; :\; x\; =\; (x\_,\; \backslash dots,\; x\_)\; \backslash mapsto\; \backslash hat\_\; =\; (x\_,\; \backslash dots,\; x\_,\; x\_,\; \backslash dots,\; x\_).$ For each 1 ≤ ''j'' ≤ ''d'', let :$g\_\; :\; \backslash mathbb^\; \backslash to\; [0,\; +\; \backslash infty),$ :$g\_\; \backslash in\; L^\; (\backslash mathbb^).$ Then the Loomis–Whitney inequality holds: :$\backslash int\_\}\; \backslash prod\_^\; g\_\; (\; \backslash pi\_\; (x)\; )\; \backslash ,\; \backslash mathrm\; x\; \backslash leq\; \backslash prod\_^\; \backslash |\; g\_\; \backslash |\_^)\}.$ Equivalently, taking :$f\_\; (x)\; =\; g\_\; (x)^,$ :$\backslash int\_\}\; \backslash prod\_^\; f\_\; (\; \backslash pi\_\; (x)\; )^\; \backslash ,\; \backslash mathrm\; x\; \backslash leq\; \backslash prod\_^\; \backslash left(\; \backslash int\_\}\; f\_\; (\backslash hat\_)\; \backslash ,\; \backslash mathrm\; \backslash hat\_\; \backslash right)^.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Loomis–Whitney inequality」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.