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H. Tverberg, A generalization of Radon’s theorem , J. London Math. Soc 41
1966
Earlier work this paper cites.
J.-P. Doignon and G. Valette, Radon partitions and a new notion of independence in affine and projective spaces , Mathematika 24
1977
Earlier work this paper cites.

Similar

A. Vučić and R. T. Živaljević, Note on a conjecture of Sierksma , Discrete & Computational Geometry 9
1993
Cited alongside, same era.
S. Hell, On the number of Tverberg partitions in the prime power case , European Journal of Combinatorics 28
2007
Cited alongside, same era.

Then

M. A. Perles and M. Sigron, Strong general position , arXiv preprint arXiv:1409.2899 (2014)
Original
2014
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2015

On Tverberg partitions

White, Moshe

Understand

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 $a_1,\ldots,a_r$.