Log-concavity of Gaussian convex-hull intersection probabilities
For integers $d \ge 1$ and $n \ge d + 2$, let $X_1, \dots, X_n$ be independent standard Gaussian vectors in $\mathbb{R}^d$, and for $1 \le k \le n - 1$ let $$q_k(n,d) = \mathbb{P}\bigl(\operatorname{conv}\{X_1, \dots, X_k\} \cap \operatorname{conv}\{X_{k+1}, \dots, X_n\} \ne \varnothing\bigr),$$ so that $q_k = q_{n-k}$. Chan, Kalai, Narayanan, Ter-Saakov and White proved that $(q_k(d+2, d))_k$ is log-concave, the case in which the points almost surely have a unique Radon partition. Together with the symmetry, log-concavity would give $q_1 \le q_2 \le \dots \le q_{\lfloor n/2 \rfloor}$, which White proposed for all $n \ge d + 2$.
Problem. Is $$q_k(n,d)^2 \ge q_{k-1}(n,d)\, q_{k+1}(n,d) \qquad (2 \le k \le n - 2)$$ for all integers $d \ge 1$ and $n \ge d + 3$?
Source: Chan, Kalai, Narayanan, Ter-Saakov and White (2025), Conjecture 7.2(2), for n ≥ d + 2; its unimodality part (1) was proposed by White (2023). Stated for n ≥ d + 3 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- CO, MG
- Keywords
- AIM Open Applied Problems, Gaussian random points, log-concave sequences, Radon partitions
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},
}
@misc{aim-gaussian-radon-intersection-logconcavity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Log-concavity of Gaussian convex-hull intersection probabilities}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/c9929805cece044705abaaebe1cebddac1ba11a6/problems/598-gaussian-radon-intersection-logconcavity.md},
note = {Entry 598 at commit c992980, 2026-10-05},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00268, "Log-concavity of Gaussian convex-hull intersection probabilities", clopen.solutions.