A Hexagonal Counterexample to Log-Convexity of Fisher Information Along the Heat Flow

Unknown Date

AI Summary

This work investigates the log-convexity of Fisher information along the heat flow, with a focus on the validity of the high-dimensional Cheng–Geng conjecture. By constructing a smooth, strictly positive density function with Gaussian decay on the two-dimensional torus and employing a Gaussian envelope mapping together with explicit numerical computations, the authors provide the first counterexamples to the conjecture in dimensions two and higher, thereby disproving both the Cheng–Geng conjecture and its associated entropy power conjecture. Furthermore, they establish that the optimal one-dimensional constant is θ₁* = 1, prove the monotonicity of the optimal constants θ_d* with respect to dimension, and show that the infinite-dimensional limit θ_∞* is determined by the symbol of a specific operator 𝒟.

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 $θ_d^*$ 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.

Problem

• Fisher information
• log-convexity
• heat flow
• Cheng–Geng conjecture
• counterexample

Innovation

• Fisher information
• log-convexity
• heat flow
• counterexample
• Gaussian perturbation