A counterexample to Pietsch's metric-entropy duality conjecture at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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A counterexample to Pietsch's metric-entropy duality conjecture

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	infinite dimensions	Levels:	1
Category:	Functional analysis	Lean version:	YES! ✔
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A counterexample to metric-entropy duality. 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.

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LEVEL 1 Counterexamples to the duality conjecture for metric entropy

released 2026-09-24 | 1 theorem · 3 lemmas · 9 proofs · 5,569 words | PLAY LEVEL 1 » (pdf)

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.

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