Sierksma's conjecture on the minimum number of Tverberg partitions

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Let m(d,r)m(d,r) denote the minimum number of unordered Tverberg partitions of a set of (d+1)(r−1)+1(d+1)(r-1)+1 points in Rd\mathbb{R}^d, where a Tverberg partition of order rr is a partition into rr sets whose convex hulls have a common point. Sierksma's conjecture. The minimum number is

m(d,r)=((r−1)!)d.m(d,r)=((r-1)!)^d.

This is a long-standing conjecture concerning the extremal number of Tverberg partitions.

References

Primary source

Swee Hong Chan, Gil Kalai, Bhargav Narayanan, Natalya Ter-Saakov and Moshe White, “Unimodality for Radon partitions of random vectors”, arXiv:2507.01353 (2025).

Additional references

2 papers in this index state this conjecture (2017–2025). The statement above is taken from the most recent of them; the others are arXiv:1706.05975.

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