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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2605.02023 |
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| _version_ | 1866913084976660480 |
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| author | Kunisky, Dmitriy |
| author_facet | Kunisky, Dmitriy |
| contents | Litvak (2018) conjectured that, for any $p > 0$, the quantity $\mathbb{E}[\min_{i = 1}^n |g_i|^p]$ where $g \sim \mathcal{N}(0, Σ)$ is a centered Gaussian random vector is minimized among $n \times n$ correlation matrices $Σ$ by the Gram matrix of the regular simplex in $\mathbb{R}^{n - 1}$. We disprove this conjecture: the matrix with entries $Σ^{\mathrm{cos}}_{ij}=\cos(π(i - j) / n)$ already achieves a smaller moment for $p = 2$ and $n = 4$. We propose that $Σ^{\mathrm{cos}}$ 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 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 $g \sim \mathcal{N}(0, Σ^{\mathrm{cos}})$ is stochastically dominated by $\min_{i = 1}^n |h_i|$ for $h \sim \mathcal{N}(0, Σ)$ for any $n \times n$ correlation matrix $Σ$. Our counterexample $Σ^{\mathrm{cos}}$ was found by the AlphaEvolve AI-assisted optimization system, and we also include a brief discussion of its application to such problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02023 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture Kunisky, Dmitriy Probability Metric Geometry Litvak (2018) conjectured that, for any $p > 0$, the quantity $\mathbb{E}[\min_{i = 1}^n |g_i|^p]$ where $g \sim \mathcal{N}(0, Σ)$ is a centered Gaussian random vector is minimized among $n \times n$ correlation matrices $Σ$ by the Gram matrix of the regular simplex in $\mathbb{R}^{n - 1}$. We disprove this conjecture: the matrix with entries $Σ^{\mathrm{cos}}_{ij}=\cos(π(i - j) / n)$ already achieves a smaller moment for $p = 2$ and $n = 4$. We propose that $Σ^{\mathrm{cos}}$ 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 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 $g \sim \mathcal{N}(0, Σ^{\mathrm{cos}})$ is stochastically dominated by $\min_{i = 1}^n |h_i|$ for $h \sim \mathcal{N}(0, Σ)$ for any $n \times n$ correlation matrix $Σ$. Our counterexample $Σ^{\mathrm{cos}}$ was found by the AlphaEvolve AI-assisted optimization system, and we also include a brief discussion of its application to such problems. |
| title | A revision of Litvak's conjecture on Gaussian minima and a volumetric zone conjecture |
| topic | Probability Metric Geometry |
| url | https://arxiv.org/abs/2605.02023 |