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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 Equilateral dimension ： ウィキペディア英語版 Equilateral dimension In mathematics, the equilateral dimension of a metric space is the maximum number of points that are all at equal distances from each other. Equilateral dimension has also been called "metric dimension", but the term "metric dimension" also has many other inequivalent usages.〔 The equilateral dimension of a ''d''-dimensional Euclidean space is , and the equilateral dimension of a ''d''-dimensional vector space with the Chebyshev distance (L∞ norm) is 2''d''. However, the equilateral dimension of a space with the Manhattan distance (L1 norm) is not known; Kusner's conjecture, named after Robert B. Kusner, states that it is exactly 2''d''.〔; .〕 ==Lebesgue spaces== The equilateral dimension has been particularly studied for Lebesgue spaces, finite-dimensional normed vector spaces with the L''p'' norm :$\backslash \; \backslash |x\backslash |\_p=\backslash left(|x\_1|^p+|x\_2|^p+\backslash cdots+|x\_d|^p\backslash right)^.$ The equilateral dimension of L''p'' spaces of dimension ''d'' behaves differently depending on the value of ''p'': *For ''p'' = 1, the L''p'' norm gives rise to Manhattan distance. In this case, it is possible to find 2''d'' equidistant points, the vertices of an axis-aligned cross polytope. The equilateral dimension is known to be exactly 2''d'' for ,〔; .〕 and to be upper bounded by for any ''d''.〔.〕 Robert B. Kusner suggested in 1983 that the equilateral dimension for this case should be exactly 2''d'';〔 this suggestion (together with a related suggestion for the equilateral dimension when ''p'' > 2) has come to be known as Kusner's conjecture. *For 1 < ''p'' < 2, the equilateral dimension is at least where ε is a constant that depends on ''p''.〔.〕 *For ''p'' = 2, the L''p'' norm is the familiar Euclidean distance. The equilateral dimension of ''d''-dimensional Euclidean space is : the vertices of an equilateral triangle, regular tetrahedron, or higher-dimensional regular simplex form an equilateral set, and every equilateral set must have this form.〔.〕 *For 2 < ''p'' < ∞, the equilateral dimension is at least : for instance the ''d'' basis vectors of the vector space together with another vector of the form for a suitable choice of ''x'' form an equilateral set. Kusner's conjecture states that in these cases the equilateral dimension is exactly . Kusner's conjecture has been proven for the special case that .〔 When ''p'' is an odd integer the equilateral dimension is upper bounded by .〔 *For ''p'' = ∞ (the limiting case of the ''L''''p'' norm for finite values of ''p'', in the limit as ''p'' grows to infinity) the ''L''''p'' norm becomes the Chebyshev distance, the maximum absolute value of the differences of the coordinates. For a ''d''-dimensional vector space with the Chebyshev distance, the equilateral dimension is 2''d'': the 2''d'' vertices of an axis-aligned hypercube are at equal distances from each other, and no larger equilateral set is possible.〔 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「Equilateral dimension」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.