Balanced Gaussian simplex noise stability for four or more classes
Let $q \ge 4$, $n \ge q - 1$ and $0 < \rho < 1$. Let $X$ and $Z$ be independent standard Gaussian vectors in $\mathbb{R}^n$ and put $Y = \rho X + \sqrt{1 - \rho^2}\, Z$. Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$. For a measurable partition $\mathcal{A} = (A_1, \dots, A_q)$ of $\mathbb{R}^n$ with $\gamma_n(A_i) = 1/q$ for every $i$, let $$\operatorname{Stab}_\rho(\mathcal{A}) = \sum_{i=1}^{q} \mathbb{P}(X \in A_i,\ Y \in A_i).$$ Let $u_1, \dots, u_q \in \mathbb{R}^n$ be unit vectors with $\langle u_i, u_j \rangle = -1/(q-1)$ for $i \ne j$, and let $\mathcal{S} = (S_1, \dots, S_q)$ be the standard simplex partition, $$S_i = \{x \in \mathbb{R}^n : \langle u_i, x \rangle \ge \langle u_j, x \rangle \text{ for every } j\},$$ with points on the boundaries, a null set, assigned arbitrarily.
Borell proved that half-spaces maximize noise stability among sets of prescribed Gaussian measure, which gives the case $q = 2$. Isaksson and Mossel conjectured the inequality below for every $q \ge 3$. Heilman, Mossel and Neeman showed that for every $q \ge 3$ and every $\rho \ne 0$ the standard simplex partition is not the most noise stable when the prescribed measures are not all equal to $1/q$. Heilman and Tarter proved that for $0 < \rho < 1$ and $q \le n + 1$ every partition of $\mathbb{R}^n$ into $q$ sets of prescribed Gaussian measures that maximizes noise stability is, after a rotation and changes on null sets, the product of a partition of $\mathbb{R}^{q-1}$ with $\mathbb{R}^{n-q+1}$. For $q = 3$ the inequality was proved for small $\rho > 0$ by Heilman, with a threshold depending on $n$, and by Heilman and Tarter with a threshold independent of $n$, and for $0 < \rho \le 2/5$ in a 2026 preprint of Heilman.
Problem. Does every such partition $\mathcal{A}$ satisfy $$\operatorname{Stab}_\rho(\mathcal{A}) \le \operatorname{Stab}_\rho(\mathcal{S})?$$
Source: Isaksson and Mossel (2012), Conjecture 1.4(i), the standard simplex conjecture, made by Mossel around 2004, for q ≥ 3. Stated for q ≥ 4 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, Gaussian partitions, noise stability
References
@article{isaksson2012,
author = {Isaksson, Marcus and Mossel, Elchanan},
title = {Maximally stable {G}aussian partitions with discrete applications},
journal = {Israel J. Math.},
volume = {189},
number = {1},
year = {2012},
pages = {347--396},
doi = {10.1007/s11856-011-0181-7},
eprint = {0903.3362},
archivePrefix = {arXiv},
}
@article{borell1985,
author = {Borell, Christer},
title = {Geometric bounds on the {O}rnstein--{U}hlenbeck velocity process},
journal = {Z. Wahrsch. Verw. Gebiete},
volume = {70},
number = {1},
year = {1985},
pages = {1--13},
doi = {10.1007/BF00532234},
}
@article{heilman2016,
author = {Heilman, Steven and Mossel, Elchanan and Neeman, Joe},
title = {Standard simplices and pluralities are not the most noise stable},
journal = {Israel J. Math.},
volume = {213},
number = {1},
year = {2016},
pages = {33--53},
doi = {10.1007/s11856-016-1320-y},
eprint = {1403.0885},
archivePrefix = {arXiv},
}
@article{heilman2014,
author = {Heilman, Steven},
title = {Euclidean partitions optimizing noise stability},
journal = {Electron. J. Probab.},
volume = {19},
year = {2014},
pages = {1--37},
doi = {10.1214/EJP.v19-3083},
eprint = {1211.7138},
archivePrefix = {arXiv},
}
@article{heilman2021,
author = {Heilman, Steven and Tarter, Alex},
title = {Three candidate plurality is stablest for small correlations},
journal = {Forum Math. Sigma},
volume = {9},
year = {2021},
pages = {e65},
doi = {10.1017/fms.2021.56},
eprint = {2011.05583},
archivePrefix = {arXiv},
}
@misc{heilman2026,
author = {Heilman, Steven},
title = {Sharp hardness for {MAX}-3-{CUT} and quantum {MAX}-{CUT}},
year = {2026},
eprint = {2608.00333},
archivePrefix = {arXiv},
}
@misc{aim-gaussian-simplex,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Balanced Gaussian simplex noise stability for four or more classes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/294-gaussian-simplex.md},
note = {Entry 294 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-00436, "Balanced Gaussian simplex noise stability for four or more classes", clopen.solutions.