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In mathematics, particularly, in asymptotic convex geometry, Milman's reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides a reverse inequality to the famous Brunn–Minkowski inequality for convex bodies in ''n''-dimensional Euclidean space R''n''. Namely, it bounds the volume of the Minkowski sum of two bodies from above in terms of the volumes of the bodies. ==Introduction== Let ''K'' and ''L'' be convex bodies in R''n''. The Brunn–Minkowski inequality states that : where vol denotes ''n''-dimensional Lebesgue measure and the + on the left-hand side denotes Minkowski addition. In general, no reverse bound is possible, since one can find convex bodies ''K'' and ''L'' of unit volume so that the volume of their Minkowski sum is arbitrarily large. Milman's theorem states that one can replace one of the bodies by its image under a properly chosen volume-preserving linear map so that the left-hand side of the Brunn–Minkowski inequality is bounded by a constant multiple of the right-hand side. The result is one of the main structural theorems in the local theory of Banach spaces. 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「Milman's reverse Brunn–Minkowski inequality」の詳細全文を読む スポンサード リンク
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