Abstract: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 $\theta_d^*$ by proving $\theta_1^*=1$, establishing monotonicity in the dimension, and identifying a dichotomy for the asymptotic constant $\theta_\infty^*$ governed by the sign of $\mathcal{D}$. The explicit two-dimensional counterexample was found by GPT-5.5 Pro.
From: Jiayang Zou [view email]
[v1]
Mon, 18 May 2026 08:58:32 UTC (22 KB)
[v2]
Tue, 19 May 2026 06:07:47 UTC (22 KB)
[v3]
Mon, 31 Aug 2026 05:46:18 UTC (25 KB)
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