Log-Convexity of Fisher Information Along Heat Flow

For every smooth positive density $f$ on $\mathbb{R}^d$ , must the Fisher information $t \mapsto I(f * \gamma_t)$ be log-convex along the heat flow?

Result: Disproved
Status: Resolved
AI contribution: AI-discovered
Method: Construction
Field: Information theory
Posed by: —
Year posed: 2015
Years open: 11y
Solved: 2026-05-18
Model: GPT-5.5 Pro
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Preprint
Significance: 15 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What the AI did

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.

Verification

Public arXiv preprint with an explicit construction and numerics. Not yet peer-reviewed.

Source

arXiv:2605.18081 - A hexagonal counterexample to log-convexity of Fisher information

Log-Convexity of Fisher Information Along Heat Flow

What the AI did

Verification

Source

Discussion