Sierksma's lower bound for Tverberg partitions
Let $d \ge 2$ and $r \ge 3$ be integers, and let $X \subset \mathbb{R}^d$ consist of exactly $(d+1)(r-1)+1$ distinct points, not necessarily in general position. A Tverberg partition of $X$ is an unordered partition $X = X_1 \sqcup \cdots \sqcup X_r$ into nonempty sets with $$\bigcap_{i=1}^r \operatorname{conv}(X_i) \ne \varnothing;$$ by Tverberg's theorem at least one exists. The number $((r-1)!)^d$ is attained, for instance by placing $r-1$ points near each vertex of a $d$-simplex and one at its barycentre; White, and Bukh, Loh and Nivasch, constructed further sets with exactly $((r-1)!)^d$ Tverberg partitions. Vučić and Živaljević proved that there are always at least $\frac{1}{(r-1)!}(r/2)^{(r-1)(d+1)/2}$ Tverberg partitions when $r$ is prime, and Hell extended this to prime powers. Hell proved that any seven points in the plane have at least four Tverberg partitions into three parts, the case $d = 2$, $r = 3$; Soberón gave an elementary proof.
Problem. Does every such $X$ have at least $((r-1)!)^d$ distinct Tverberg partitions? Partitions that differ only in the order of their parts are counted once.
Source: Sierksma (1979), in mimeographed notes (the Dutch cheese problem); see Bárány and Soberón (2018), Conjecture 4.1, and Chan, Kalai, Narayanan, Ter-Saakov and White (2026), Conjecture 7.3. Stated for d ≥ 2 and r ≥ 3 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- AT, CO
- Keywords
- AIM Open Applied Problems, Sierksma's conjecture, Tverberg partitions
References
@unpublished{sierksma1979,
author = {Sierksma, Gerard},
title = {Convexity without linearity; the {D}utch cheese problem},
note = {Mimeographed notes},
year = {1979},
}
@article{barany2018,
author = {B{\'a}r{\'a}ny, Imre and Sober{\'o}n, Pablo},
title = {Tverberg's theorem is 50 years old: a survey},
journal = {Bull. Amer. Math. Soc. (N.S.)},
volume = {55},
number = {4},
year = {2018},
pages = {459--492},
doi = {10.1090/bull/1634},
eprint = {1712.06119},
archivePrefix = {arXiv},
}
@article{chan2026,
author = {Chan, Swee Hong and Kalai, Gil and Narayanan, Bhargav and Ter-Saakov, Natalya and White, Moshe},
title = {Unimodality for {R}adon partitions of random vectors},
journal = {Discrete Comput. Geom.},
year = {2026},
doi = {10.1007/s00454-026-00870-6},
eprint = {2507.01353},
archivePrefix = {arXiv},
}
@article{tverberg1966,
author = {Tverberg, H.},
title = {A generalization of {R}adon's theorem},
journal = {J. London Math. Soc.},
volume = {41},
number = {1},
year = {1966},
pages = {123--128},
doi = {10.1112/jlms/s1-41.1.123},
}
@article{white2017,
author = {White, Moshe J.},
title = {On {T}verberg partitions},
journal = {Israel J. Math.},
volume = {219},
number = {2},
year = {2017},
pages = {549--553},
doi = {10.1007/s11856-017-1490-2},
}
@article{bukh2017,
author = {Bukh, Boris and Loh, Po-Shen and Nivasch, Gabriel},
title = {Classifying unavoidable {T}verberg partitions},
journal = {J. Comput. Geom.},
volume = {8},
number = {1},
year = {2017},
pages = {174--205},
doi = {10.20382/jocg.v8i1a9},
eprint = {1611.01078},
archivePrefix = {arXiv},
}
@article{vucic1993,
author = {Vu{\v{c}}i{\'c}, Aleksandar and {\v{Z}}ivaljevi{\'c}, Rade T.},
title = {Note on a conjecture of {S}ierksma},
journal = {Discrete Comput. Geom.},
volume = {9},
number = {4},
year = {1993},
pages = {339--349},
doi = {10.1007/BF02189327},
}
@article{hell2007,
author = {Hell, Stephan},
title = {On the number of {T}verberg partitions in the prime power case},
journal = {European J. Combin.},
volume = {28},
number = {1},
year = {2007},
pages = {347--355},
doi = {10.1016/j.ejc.2005.06.005},
eprint = {math/0404406},
archivePrefix = {arXiv},
}
@article{hell2008,
author = {Hell, Stephan},
title = {Tverberg's theorem with constraints},
journal = {J. Combin. Theory Ser. A},
volume = {115},
number = {8},
year = {2008},
pages = {1402--1416},
doi = {10.1016/j.jcta.2008.02.007},
eprint = {0704.2713},
archivePrefix = {arXiv},
}
@misc{soberon2026,
author = {Sober{\'o}n, Pablo},
title = {An elementary proof of {S}ierksma's conjecture for seven points in the plane},
year = {2026},
eprint = {2604.18485},
archivePrefix = {arXiv},
}
@misc{aim-sierksma-tverberg-partition-count,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sierksma's lower bound for Tverberg partitions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/600-sierksma-tverberg-partition-count.md},
note = {Entry 600 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00898, "Sierksma's lower bound for Tverberg partitions", clopen.solutions.