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 $\Sigma^{\mathrm{cos}}_{ij}=\cos(\pi(i - j) / n)$ already achieves a smaller moment for $p = 2$ and $n = 4$. We propose that $\Sigma^{\mathrm{cos}}$ is in fact the correct minimizer of these moments for all $p > 0$ and $n \geq 1$. Towards proving this, we conjecture a volumetric extension of Fejes T\'{o}th'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 $g \sim \mathcal{N}(0, \Sigma^{\mathrm{cos}})$ 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 $\Sigma^{\mathrm{cos}}$ was found by the AlphaEvolve AI-assisted optimization system, and we also include a brief discussion of its application to such problems.