A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture

Abstract

Litvak (2018) conjectured that, for any $p>0$, the quantity $\mathbb{E}[{\min }_{i=1}^n|g_i{|}^p]$ where $g\sim \mathcal{N}(0,\Sigma )$ is a centered Gaussian random vector is minimized among $n\times n$ correlation matrices $\Sigma$ by the Gram matrix of the regular simplex in ${\mathbb{R}}^{n-1}$. We disprove this conjecture: the matrix with entries ${\Sigma }_{ij}^{\cos }=\cos (\pi (i-j)/n)$ already achieves a smaller moment for $p=2$ and $n=4$. We propose that ${\Sigma }^{\cos }$ is in fact the correct minimizer of these moments for all $p>0$ and $n\ge 1$. Towards proving this, we conjecture a volumetric extension of Fejes Tóth's zone conjecture (1973), whose covering version was proved by Jiang and Polyanskii (2017). Conditional on this conjecture, we show the stronger result that ${\min }_{i=1}^n|g_i|$ for $g\sim \mathcal{N}(0,{\Sigma }^{\cos })$ is stochastically dominated by ${\min }_{i=1}^n|h_i|$ for $h\sim \mathcal{N}(0,\Sigma )$ for any $n\times n$ correlation matrix $\Sigma$. Our counterexample ${\Sigma }^{\cos }$ was found by the AlphaEvolve AI-assisted optimization system, and we also include a brief discussion of its application to such problems.

10 pages, 3 figures