Functional analysis·OpenAI Math·Result 329·Lean formalisation listed
A counterexample to metric-entropy duality
One of the 372 results in OpenAI's public mathematics catalogue, filed under functional analysis: 1 preprint, the latest dated 24 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and will embed the animation.
Latest paper
Counterexamples to the duality conjecture for metric entropy
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What OpenAI claims
Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube. (Lean)
Paper
Counterexamples to the duality conjecture for metric entropy
We disprove Pietsch's dimension-free duality conjecture for metric entropy. For every proposed pair of universal constants, we construct origin-symmetric convex bodies that violate the corresponding covering-entropy inequality, already when the covering body is a cube.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
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