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)
Manuscript
Counterexamples to the duality conjecture for metric entropy
September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026}
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.
Result 329 · Functional analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md