A Hexagonal Counterexample to Log-Convexity of Fisher Information Along the Heat Flow
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911695829467136 |
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| author | Zou, Jiayang Fan, Luyao Gao, Jiayang Wang, Jia |
| author_facet | Zou, Jiayang Fan, Luyao Gao, Jiayang Wang, Jia |
| contents | We construct a smooth, strictly positive, Gaussian-decaying density on $\mathbb{R}^2$ for which Fisher information along the heat flow is not log-convex. This disproves the Cheng--Geng log-convexity conjecture in dimension two and, by tensorization, in every dimension $d\ge2$. Consequently, the multidimensional forms of the Gaussian completely monotone conjecture, McKean's conjecture, and Toscani's entropy power conjecture also fail, complementing the one-dimensional counterexample of Gu and Sellke. Our construction is a small hexagonal perturbation on the triangular torus, transferred to $\mathbb{R}^2$ by a Gaussian envelope and supported by explicit two-dimensional numerics. We also initiate the study of the sharp constants $θ_d^*$ by proving $θ_1^*=1$, establishing monotonicity in the dimension, and identifying a dichotomy for the asymptotic constant $θ_\infty^*$ governed by the sign of $\mathcal{D}$. The explicit two-dimensional counterexample was found by GPT-5.5 Pro. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18081 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Hexagonal Counterexample to Log-Convexity of Fisher Information Along the Heat Flow Zou, Jiayang Fan, Luyao Gao, Jiayang Wang, Jia Information Theory We construct a smooth, strictly positive, Gaussian-decaying density on $\mathbb{R}^2$ for which Fisher information along the heat flow is not log-convex. This disproves the Cheng--Geng log-convexity conjecture in dimension two and, by tensorization, in every dimension $d\ge2$. Consequently, the multidimensional forms of the Gaussian completely monotone conjecture, McKean's conjecture, and Toscani's entropy power conjecture also fail, complementing the one-dimensional counterexample of Gu and Sellke. Our construction is a small hexagonal perturbation on the triangular torus, transferred to $\mathbb{R}^2$ by a Gaussian envelope and supported by explicit two-dimensional numerics. We also initiate the study of the sharp constants $θ_d^*$ by proving $θ_1^*=1$, establishing monotonicity in the dimension, and identifying a dichotomy for the asymptotic constant $θ_\infty^*$ governed by the sign of $\mathcal{D}$. The explicit two-dimensional counterexample was found by GPT-5.5 Pro. |
| title | A Hexagonal Counterexample to Log-Convexity of Fisher Information Along the Heat Flow |
| topic | Information Theory |
| url | https://arxiv.org/abs/2605.18081 |