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arXiv · 1508.07262

On Tverberg partitions

Moshe White
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Abstract

A theorem of Tverberg from 1966 asserts that every set $X\subset\mathbb{R}^d$ 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 $a_1,\ldots,a_r$ satisfying $n=a_1+\cdots+a_r$), where the parts $a_i$ correspond to the number of points in every subset. In this paper, we prove that for any partition $a_i\le d+1$, $i=1,\ldots,r$, there exists a set $X\subset\mathbb{R}^d$ of $n$ points, such that every Tverberg partition of $X$ induces the same partition on $n$, given by the parts $a_1,\ldots,a_r$.

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Moshe White. 2017-05-15. On Tverberg partitions. https://doi.org/10.1007/s11856-017-1490-2

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