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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 Radon's theorem ： ウィキペディア英語版 Radon's theorem In geometry, Radon's theorem on convex sets, named after Johann Radon, states that any set of ''d'' + 2 points in R''d'' can be partitioned into two disjoint sets whose convex hulls intersect. A point in the intersection of these convex hulls is called a Radon point of the set. For example, in the case ''d'' = 2, any set of four points in the Euclidean plane can be partitioned in one of two ways. It may form a triple and a singleton, where the convex hull of the triple (a triangle) contains the singleton; alternatively, it may form two pairs of points that form the endpoints of two intersecting line segments. ==Proof and construction== Consider any set $\backslash \backslash subset\; \backslash mathbf^d$ of ''d'' + 2 points in ''d''-dimensional space. Then there exists a set of multipliers ''a''1, ..., ''a''''d'' + 2, not all of which are zero, solving the system of linear equations :$\backslash sum\_^\; a\_i\; x\_i=0,\backslash quad\; \backslash sum\_^\; a\_i=0,$ because there are ''d'' + 2 unknowns (the multipliers) but only ''d'' + 1 equations that they must satisfy (one for each coordinate of the points, together with a final equation requiring the sum of the multipliers to be zero). Fix some particular nonzero solution ''a''1, ..., ''a''''d'' + 2. Let ''I'' be the set of points with positive multipliers, and let ''J'' be the set of points with multipliers that are negative or zero. Then ''I'' and ''J'' form the required partition of the points into two subsets with intersecting convex hulls. The convex hulls of ''I'' and ''J'' must intersect, because they both contain the point :$p=\; \backslash sum\_\backslash frac\; x\_i=\backslash sum\_\backslash fracx\_j,$ where :$A=\backslash sum\_\; a\_i=-\backslash sum\_\; a\_j.$ The left hand side of the formula for ''p'' expresses this point as a convex combination of the points in ''I'', and the right hand side expresses it as a convex combination of the points in ''J''. Therefore, ''p'' belongs to both convex hulls, completing the proof. This proof method allows for the efficient construction of a Radon point, in an amount of time that is polynomial in the dimension, by using Gaussian elimination or other efficient algorithms to solve the system of equations for the multipliers.〔.〕 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「Radon's theorem」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.