A proof of the strong Gaussian product inequality conjecture
Abstract
Let X = (X1,…,Xn) be a centered Gaussian vector, not necessarily nondegenerate. It is proved that, for every α1,…,αn > 0, E[∏i=1n |Xi|αi] ≥ ∏i=1n E[|Xi|αi]. When all marginal variances are positive, equality holds if and only if the coordinates are independent. This settles Frenkel's 18-year-old Gaussian product inequality (GPI) conjecture and, in fact, its later strengthening to arbitrary positive exponents. Through Frenkel's hafnian formulation, this result also provides a short proof of the 28-year-old real linear polarization constant conjecture, which was settled this year as a consequence of the strong polarization inequality. The main result leads to the exact real linear polarization constant, a sharp weighted product inequality for real linear functionals, spherical moment bounds, hafnian inequalities for positive-semidefinite matrices together with complete characterizations of their equality cases, logarithmic variance and covariance inequalities, a Rényi total-correlation certificate, and unconditional versions of mixed-sign Gaussian moment bounds previously conditional on the positive-exponent GPI.
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BibTeX
@misc{ouimet2026proof,
title = {{A proof of the strong Gaussian product inequality conjecture}},
author = {Frédéric Ouimet and Dylan Greaves},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20234},
eprint = {2609.20234},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20234},
}
RIS
TY - UNPB
TI - A proof of the strong Gaussian product inequality conjecture
AU - Frédéric Ouimet
AU - Dylan Greaves
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20234
UR - https://arxiv.org/abs/2609.20234
AB - Let X = (X_1,…,X_n) be a centered Gaussian vector, not necessarily nondegenerate. It is proved that, for every α_1,…,α_n > 0, E[∏_i=1^n |X_i|^α_i] ≥ ∏_i=1^n E[|X_i|^α_i]. When all marginal variances are positive, equality holds if and only if the coordinates are independent. This settles Frenkel's 18-year-old Gaussian product inequality (GPI) conjecture and, in fact, its later strengthening to arbitrary positive exponents. Through Frenkel's hafnian formulation, this result also provides a short proof of the 28-year-old real linear polarization constant conjecture, which was settled this year as a consequence of the strong polarization inequality. The main result leads to the exact real linear polarization constant, a sharp weighted product inequality for real linear functionals, spherical moment bounds, hafnian inequalities for positive-semidefinite matrices together with complete characterizations of their equality cases, logarithmic variance and covariance inequalities, a Rényi total-correlation certificate, and unconditional versions of mixed-sign Gaussian moment bounds previously conditional on the positive-exponent GPI.
ER -