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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 Borsuk's conjecture ： ウィキペディア英語版 Borsuk's conjecture The Borsuk problem in geometry, for historical reasons incorrectly called Borsuk's conjecture, is a question in discrete geometry. ==Problem== In 1932 Karol Borsuk showed that an ordinary 3-dimensional ball in Euclidean space can be easily dissected into 4 solids, each of which has a smaller diameter than the ball, and generally ''d''-dimensional ball can be covered with compact sets of diameters smaller than the ball. At the same time he proved that ''d'' subsets are not enough in general. The proof is based on the Borsuk–Ulam theorem. That led Borsuk to a general question: : ''Die folgende Frage bleibt offen: Lässt sich jede beschränkte Teilmenge E des Raumes $\backslash Bbb\; R^n$ in (n + 1) Mengen zerlegen, von denen jede einen kleineren Durchmesser als E hat?''〔 This can be translated as: : ''The following question remains open: Can every bounded subset E of the space $\backslash Bbb\; R^n$ be partitioned into (n + 1) sets, each of which has a smaller diameter than E?'' The question got a positive answer in the following cases: * ''d'' = 2 — which is the original result by Karol Borsuk (1932). * ''d'' = 3 — shown by Julian Perkal (1947), and independently, 8 years later, by H. G. Eggleston (1955). A simple proof was found later by Branko Grünbaum and Aladár Heppes. * For all ''d'' for the smooth convex bodies — shown by Hugo Hadwiger (1946). * For all ''d'' for centrally-symmetric bodies — shown by A.S. Riesling (1971). * For all ''d'' for bodies of revolution — shown by Boris Dekster (1995). The problem was finally solved in 1993 by Jeff Kahn and Gil Kalai, who showed that the general answer to Borsuk's question is ''no''. Their construction shows that pieces do not suffice for and for each . After Andriy V. Bondarenko had shown that Borsuk’s conjecture is false for all , the current best bound, due to Thomas Jenrich, is 64. Apart from finding the minimum number ''d'' of dimensions such that the number of pieces $\backslash alpha(d)\; >\; d+1$ mathematicians are interested in finding the general behavior of the function $\backslash alpha(d)$. Kahn and Kalai show that in general (that is for ''d'' big enough), one needs $\backslash alpha(d)\; \backslash ge\; (1.2)^\backslash sqrt$ number of pieces. They also quote the upper bound by Oded Schramm, who showed that for every ''ε'', if ''d'' is sufficiently large, $\backslash alpha(d)\; \backslash le\; \backslash left(\backslash sqrt\; +\; \backslash varepsilon\backslash right)^d$. The correct order of magnitude of ''α''(''d'') is still unknown. However, it is conjectured that there is a constant such that $\backslash alpha(d)\; >\; c^d$ for all . 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「Borsuk's conjecture」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.