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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 Vitale's random Brunn–Minkowski inequality ： ウィキペディア英語版 Vitale's random Brunn–Minkowski inequality In mathematics, Vitale's random Brunn–Minkowski inequality is a theorem due to Richard Vitale that generalizes the classical Brunn–Minkowski inequality for compact subsets of ''n''-dimensional Euclidean space R''n'' to random compact sets. ==Statement of the inequality== Let ''X'' be a random compact set in R''n''; that is, a Borel–measurable function from some probability space (Ω, Σ, Pr) to the space of non-empty, compact subsets of R''n'' equipped with the Hausdorff metric. A random vector ''V'' : Ω → R''n'' is called a selection of ''X'' if Pr(''V'' ∈ ''X'') = 1. If ''K'' is a non-empty, compact subset of R''n'', let :$\backslash |\; K\; \backslash |\; =\; \backslash max\; \backslash left\backslash \}\; \backslash right|\; v\; \backslash in\; K\; \backslash right\backslash \}$ and define the expectation E() of ''X'' to be : Note that E() is a subset of R''n''. In this notation, Vitale's random Brunn–Minkowski inequality is that, for any random compact set ''X'' with E() < +∞, :$\backslash left(\; \backslash mathrm\; \backslash left(\; \backslash mathrm\; ()\; \backslash right)\; \backslash right)^\; \backslash geq\; \backslash mathrm\; \backslash left(\backslash mathrm\; (X)^\; \backslash right),$ where "vol" denotes ''n''-dimensional Lebesgue measure. 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「Vitale's random Brunn–Minkowski 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.