Unimodality of Radon partitions for uniform convex samples
Let $d \ge 2$, let $K \subset \mathbb{R}^d$ be a convex body (compact, with nonempty interior), set $n = d + 2$, and let $X_1, \dots, X_n$ be independent and uniformly distributed on $K$. Almost surely the points have a unique Radon partition $\{A, B\}$: a partition into two nonempty sets whose convex hulls intersect. Choose one of $A$ and $B$ with probability $1/2$ each, independently of the sample, and let $p_k$ be the probability that the chosen set has $k$ elements, so that $p_k = p_{n-k}$. For independent standard Gaussian points in place of uniform ones, Chan, Kalai, Narayanan, Ter-Saakov and White proved that $(p_0, \dots, p_n)$ is ultra log-concave, and hence unimodal, which White had conjectured. For $d = 2$ the two sets have two points each exactly when the four points are in convex position; Blaschke proved that four independent uniform points in a planar convex body are in convex position with probability at least $2/3$, with equality for triangles.
Problem. Is $p_1 \le p_2 \le \dots \le p_{\lfloor n/2 \rfloor}$ for every $d \ge 2$ and every convex body $K \subset \mathbb{R}^d$?
Source: Chan, Kalai, Narayanan, Ter-Saakov and White (2026), Problem 7.1 and Section 1, who ask whether their theorem for Gaussian points, or its unimodality corollary, extends to points drawn uniformly from a fixed convex set; stated for unimodality by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- CO, MG
- Keywords
- AIM Open Applied Problems, Radon partitions, Sylvester's four-point problem, unimodal sequences
References
@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},
}
@phdthesis{white2023,
author = {White, Moshe},
title = {Radon partitions of random {G}aussian polytopes},
school = {The Hebrew University of Jerusalem},
year = {2023},
}
@article{blaschke1917,
author = {Blaschke, Wilhelm},
title = {{\"U}ber affine {G}eometrie {XI}: {L}{\"o}sung des {V}ierpunktproblems von {S}ylvester aus der {T}heorie der geometrischen {W}ahrscheinlichkeiten},
journal = {Ber. Verh. S{\"a}chs. Ges. Wiss. Leipzig Math.-Phys. Kl.},
volume = {69},
year = {1917},
pages = {436--453},
}
@misc{aim-uniform-convex-radon-unimodality,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Unimodality of Radon partitions for uniform convex samples}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/599-uniform-convex-radon-unimodality.md},
note = {Entry 599 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00312, "Unimodality of Radon partitions for uniform convex samples", clopen.solutions.