On Tverberg partitions |
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Authors: | Moshe J White |
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Institution: | 1.Institute of Mathematics,The Hebrew University of Jerusalem,Givat Ram, Jerusalem,Israel |
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Abstract: | A theorem of Tverberg from 1966 asserts that every set X ? ? d of n = T(d, r) = (d + 1)(r ? 1) + 1 points can be partitioned into r pairwise disjoint subsets, whose convex hulls have a point in common. Thus every such partition induces an integer partition of n into r parts (that is, r integers a 1,..., a r satisfying n = a 1 + ··· + a r ), in which the parts a i correspond to the number of points in every subset. In this paper, we prove that for any partition of n where the parts satisfy a i ≤ d + 1 for all i = 1,..., r, there exists a set X ? ? of n points, such that every Tverberg partition of X induces the same partition on n, given by the parts a 1,..., a r . |
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