Log-Convexity of Fisher Information Along Heat Flow
Statement
For every smooth positive density on , must the Fisher information be log-convex along the heat flow?
Context
A known question in the information-theoretic entropy-power circle.
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The hexagonal counterexample - a smooth positive Gaussian-decaying density on the plane - was found with GPT-5.5 Pro; tensorization extends the disproof to every dimension at least two.
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