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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 Doubling space ： ウィキペディア英語版 Doubling space In mathematics, a metric space ''X'' with metric ''d'' is said to be doubling if there is some constant ''M'' > 0 such that for any ''x'' in ''X'' and ''r'' > 0, the ball ''B''(''x'', ''r'') =  may be contained in a union of no more than ''M'' many balls with radius ''r''/2. The base-2 logarithm of ''M'' is often referred to as the doubling dimension of ''X''. Euclidean spaces ℝ''d'' equipped with the usual Euclidean metric are examples of doubling spaces where the doubling constant ''M'' depends on the dimension ''d''. ==Assouad's embedding theorem== An important question in metric space geometry is to characterize those metric spaces that can be embedded in some Euclidean space by a bi-Lipschitz function. This means that one can essentially think of the metric space as a subset of Euclidean space. Not all metric spaces may be embedded in Euclidean space. Doubling metric spaces, on the other hand, would seem like they have more of a chance, since the doubling condition says, in a way, that the metric space is not infinite dimensional. However, this is still not the case in general. The Heisenberg group with its Carnot metric is an example of a doubling metric space which cannot be embedded in any Euclidean space. Assouad's Theorem states that, for a ''M''-doubling metric space ''X'', if we give it the metric ''d''(''x'', ''y'')''ε'' for some 0 < ''ε'' < 1, then there is a ''L''-bi-Lipschitz map ''f'':''X'' → ℝ''d'', where ''d'' and ''L'' depend on ''M'' and ''ε''. 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「Doubling space」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.