翻訳と辞書 Words near each other ・ Carathis ・ Carathis alayorum ・ Carathis australis ・ Carathis byblis ・ Carathis gortynoides ・ Carathis palpalis ・ Carathis septentrionalis ・ Carathéodory conjecture ・ Carathéodory kernel theorem ・ Carathéodory metric ・ Carathéodory's criterion ・ Carathéodory's existence theorem ・ Carathéodory's extension theorem ・ Carathéodory's theorem ・ Carathéodory's theorem (conformal mapping) ・ Carathéodory's theorem (convex hull) ・ Carathéodory–Jacobi–Lie theorem ・ Caratinga ・ CaratLane ・ Caratoola Recreation Park ・ Caratti ・ Caratunk Falls Archeological District ・ Caratunk, Maine ・ Carau ・ Carauari ・ Carauari Airport ・ Caraula ・ Caraula River ・ Caraulun River ・ Caraun Reid Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Carathéodory's theorem (convex hull) ： ウィキペディア英語版 Carathéodory's theorem (convex hull) :''See also Carathéodory's theorem (disambiguation) for other meanings'' In convex geometry Carathéodory's theorem states that if a point ''x'' of R''d'' lies in the convex hull of a set ''P'', there is a subset ′ of ''P'' consisting of ''d'' + 1 or fewer points such that ''x'' lies in the convex hull of ′. Equivalently, ''x'' lies in an ''r''-simplex with vertices in ''P'', where $r\; \backslash leq\; d$. The result is named for Constantin Carathéodory, who proved the theorem in 1911 for the case when ''P'' is compact. In 1914 Ernst Steinitz expanded Carathéodory's theorem for any sets ''P'' in Rd. For example, consider a set ''P'' = which is a subset of R2. The convex hull of this set is a square. Consider now a point ''x'' = (1/4, 1/4), which is in the convex hull of ''P''. We can then construct a set = ′, the convex hull of which is a triangle and encloses ''x'', and thus the theorem works for this instance, since |′| = 3. It may help to visualise Carathéodory's theorem in 2 dimensions, as saying that we can construct a triangle consisting of points from ''P'' that encloses any point in ''P''. ==Proof== Let ''x'' be a point in the convex hull of ''P''. Then, ''x'' is a convex combination of a finite number of points in ''P'' : :$\backslash mathbf=\backslash sum\_^k\; \backslash lambda\_j\; \backslash mathbf\_j$ where every ''x''j is in ''P'', every ''λ''j is non-negative, and $\backslash sum\_^k\backslash lambda\_j=1$. Suppose ''k'' > ''d'' + 1 (otherwise, there is nothing to prove). Then, the points ''x''2 − ''x''1, ..., ''x''''k'' − ''x''1 are linearly dependent, so there are real scalars ''μ''2, ..., ''μ''''k'', not all zero, such that :$\backslash sum\_^k\; \backslash mu\_j\; (\backslash mathbf\_j-\backslash mathbf\_1)=\backslash mathbf.$ If ''μ''1 is defined as :$\backslash mu\_1:=\; -\backslash sum\_^k\; \backslash mu\_j$ then :$\backslash sum\_^k\; \backslash mu\_j\; \backslash mathbf\_j=\backslash mathbf$ :$\backslash sum\_^k\; \backslash mu\_j=0$ and not all of the μ''j'' are equal to zero. Therefore, at least one ''μ''j > 0. Then, :$\backslash mathbf\; =\; \backslash sum\_^k\; \backslash lambda\_j\; \backslash mathbf\_j-\backslash alpha\backslash sum\_^k\; \backslash mu\_j\; \backslash mathbf\_j\; =\; \backslash sum\_^k\; (\backslash lambda\_j-\backslash alpha\backslash mu\_j)\; \backslash mathbf\_j$ for any real ''α''. In particular, the equality will hold if ''α'' is defined as :$\backslash alpha:=\backslash min\_\; \backslash left\backslash :\backslash mu\_j>0\backslash right\backslash \}=\backslash tfrac.$ Note that ''α'' > 0, and for every ''j'' between 1 and ''k'', :$\backslash lambda\_j-\backslash alpha\backslash mu\_j\; \backslash geq\; 0.$ In particular, ''λ''''i'' − ''αμ''''i'' = 0 by definition of ''α''. Therefore, :$\backslash mathbf\; =\; \backslash sum\_^k\; (\backslash lambda\_j-\backslash alpha\backslash mu\_j)\; \backslash mathbf\_j$ where every $\backslash lambda\_j\; -\; \backslash alpha\; \backslash mu\_j$ is nonnegative, their sum is one , and furthermore, $\backslash lambda\_i-\backslash alpha\backslash mu\_i=0$. In other words, ''x'' is represented as a convex combination of at most ''k''-1 points of ''P''. This process can be repeated until ''x'' is represented as a convex combination of at most ''d'' + 1 points in ''P''. An alternative proof uses Helly's theorem. 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「Carathéodory's theorem (convex hull)」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.