A hexagonal perturbation on the triangular torus, wrapped in a Gaussian envelope, yields an explicit two-dimensional counterexample to log-convexity of Fisher information along the heat flow, disproving the Cheng-Geng conjecture for d greater than or equal to 2.
Log-convexity of Fisher information along heat flow,
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A Hexagonal Counterexample to Log-Convexity of Fisher Information Along the Heat Flow
A hexagonal perturbation on the triangular torus, wrapped in a Gaussian envelope, yields an explicit two-dimensional counterexample to log-convexity of Fisher information along the heat flow, disproving the Cheng-Geng conjecture for d greater than or equal to 2.