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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 Mahler volume ： ウィキペディア英語版 Mahler volume In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the body and is invariant under linear transformations. It is named after German-English mathematician Kurt Mahler. It is known that the shapes with the largest possible Mahler volume are the spheres and ellipsoids; this is now known as the Blaschke–Santaló inequality. The still-unsolved Mahler conjecture states that the minimum possible Mahler volume is attained by a hypercube. ==Definition== A convex body in Euclidean space is defined as a compact convex set with non-empty interior. If ''B'' is a centrally symmetric convex body in ''n''-dimensional Euclidean space, the polar body ''B''o is another centrally symmetric body in the same space, defined as the set :$\backslash left\backslash .$ The Mahler volume of ''B'' is the product of the volumes of ''B'' and ''B''o.〔.〕 If ''T'' is a linear transformation, then $(TB)^\backslash circ\; =\; (T^)^\backslash ast\; B^\backslash circ$; thus applying ''T'' to ''B'' changes its volume by $\backslash det\; T$ and changes the volume of ''B''o by $\backslash det\; (T^)^\backslash ast$. Thus the overall Mahler volume of ''B'' is preserved by linear transformations. 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「Mahler volume」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.