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

Abstract

We construct a smooth, strictly positive, Gaussian-decaying density on ℝ² 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≥2. 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 ℝ² by a Gaussian envelope and supported by explicit two-dimensional numerics. We also initiate the study of the sharp constants θ_d^* by proving θ₁^*=1, establishing monotonicity in the dimension, and identifying a dichotomy for the asymptotic constant θ_∞^* governed by the sign of 𝒟. The explicit two-dimensional counterexample was found by GPT-5.5 Pro.

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BibTeX RIS

BibTeX

@misc{zou2026hexagonal,
  title = {{A Hexagonal Counterexample to Log-Convexity of Fisher Information Along the Heat Flow}},
  author = {Jiayang Zou and Luyao Fan and Jiayang Gao and Jia Wang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2605.18081},
  eprint = {2605.18081},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.18081},
}

RIS

TY  - UNPB
TI  - A Hexagonal Counterexample to Log-Convexity of Fisher Information Along the Heat Flow
AU  - Jiayang Zou
AU  - Luyao Fan
AU  - Jiayang Gao
AU  - Jia Wang
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2605.18081
UR  - https://arxiv.org/abs/2605.18081
AB  - We construct a smooth, strictly positive, Gaussian-decaying density on 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≥2. 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 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_∞^* governed by the sign of 𝒟. The explicit two-dimensional counterexample was found by GPT-5.5 Pro.
ER  -

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