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ON TVERBERG PARTITIONS

机译：在Tverberg分区上

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摘要

A theorem of Tverberg from 1966 asserts that every set X subset of R-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) vertical bar...vertical bar 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 subset of R 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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