Stochastic Domination of Gaussian Maxima by the Regular Simplex
Abstract
Let n≥2, and let X=(X1,…,Xn) be a centered Gaussian vector with Var(Xi)=1 for every i. Let Z1,…,Zn be independent standard Gaussians, and put Z̅=(Z1+⋯+Zn)/n. We prove ℙ{maxi Xi≤ t}≥ℙ{√(n/(n−1)) maxi(Zi−Z̅)≤ t} for every t∈ℝ, and for each fixed t>0 equality holds only when Cov(Xi,Xj)=−1/(n−1) for all i≠ j. The right side is the distribution function of the maximum of the regular simplex vector. Equivalently, among all simplices containing a given centered ball, the regular simplex circumscribed about the ball has the least standard Gaussian measure, as conjectured by Balitskiy, Karasev, and Tsigler. In our preceding paper we proved this comparison after both maxima are smoothed by independent Gaussian noise of variance 1/(n−1), which suffices for the Weak Simplex Conjecture; here we remove the smoothing, which is what probabilities at a single threshold require. As an application we consider n equally likely signals of equal energy in Gaussian noise, where the transmitter may also send nothing. At every positive false-alarm level, and for every law of a common nonnegative random amplitude not concentrated at zero, the regular simplex uniquely maximizes the average probability of correct identification whenever the signal dimension is at least n−1. A Lean formalization is available at https://github.com/abhmul/full-simplex-conjecture-lean.
Related
- Linear stability and divisibility gaps for Gaussian maxima
- Minimal central slices of the regular simplex
- Local Maxima of Discrete Gaussian Processes: Exact Distributions and Infill Limits
- Stability of Hyperplane Projections of the Regular Simplex and Lp-Projection Bodies of balls in ℓqn(ℝ)
- Regular simplex tensors: optimization landscape, conjecture proof, and beyond
BibTeX
@misc{mulgund2026stochastic,
title = {{Stochastic Domination of Gaussian Maxima by the Regular Simplex}},
author = {Abhijeet Mulgund},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28452},
eprint = {2609.28452},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28452},
}
RIS
TY - UNPB
TI - Stochastic Domination of Gaussian Maxima by the Regular Simplex
AU - Abhijeet Mulgund
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.28452
UR - https://arxiv.org/abs/2609.28452
AB - Let n≥2, and let X=(X_1,…,X_n) be a centered Gaussian vector with Var(X_i)=1 for every i. Let Z_1,…,Z_n be independent standard Gaussians, and put Z̅=(Z_1+⋯+Z_n)/n. We prove ℙ{max_i X_i≤ t}≥ℙ{√(n/(n−1)) max_i(Z_i−Z̅)≤ t} for every t∈ℝ, and for each fixed t>0 equality holds only when Cov(X_i,X_j)=−1/(n−1) for all i≠ j. The right side is the distribution function of the maximum of the regular simplex vector. Equivalently, among all simplices containing a given centered ball, the regular simplex circumscribed about the ball has the least standard Gaussian measure, as conjectured by Balitskiy, Karasev, and Tsigler. In our preceding paper we proved this comparison after both maxima are smoothed by independent Gaussian noise of variance 1/(n−1), which suffices for the Weak Simplex Conjecture; here we remove the smoothing, which is what probabilities at a single threshold require. As an application we consider n equally likely signals of equal energy in Gaussian noise, where the transmitter may also send nothing. At every positive false-alarm level, and for every law of a common nonnegative random amplitude not concentrated at zero, the regular simplex uniquely maximizes the average probability of correct identification whenever the signal dimension is at least n−1. A Lean formalization is available at https://github.com/abhmul/full-simplex-conjecture-lean.
ER -