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The duality conjecture for metric entropy (Pietsch)

Theorem 1.1: for every $a,b\ge1$ there exist $n$ and an origin-symmetric convex body $K\subset\mathbb R^n$ such that, with $L$ the cube, $\log N(K,L)>b\log N(L^\circ,a^{-1}K^\circ)$ . A further construction (per the Lean scope note) makes the dual-to-primal entropy ratio tend to zero. This disproves the two-sided dimension-free conjecture. It leaves intact the ellipsoid case, the convexified-packing duality, bounds with constants depending on the K-convexity constant, and Milman's logarithmic-loss comparison; no fixed-dimension claim is made.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The duality conjecture for metric entropy (Pietsch) · Duality conjecture for metric entropy

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed
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Unreleased internal OpenAI model
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VibeMathed record: The duality conjecture for metric entropy (Pietsch)

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture: for every $a,b\ge1$ there are $n$ and an origin-symmetric body $K\subset\mathbb R^n$ with, for $L=[-1,1]^n$ , $\log N(K,L)>b\log N(L^\circ,a^{-1}K^\circ)$ , so the right-hand inequality fails for every pair of constants. lean/formalization.yaml lists MetricEntropyDuality.json (declaration OAI.MetricEntropyDuality.exists_entropy_duality_counterexample_with_covers, file OAI/Analysis/MetricEntropy/Main.lean). The statement MetricEntropyDuality.lean was read here; not rebuilt here. It defines the cube, polar, translate covers and covering numbers in $\mathbb R^n$ , and states exactly Theorem 1.1 for a compact convex symmetric $K$ with nonempty interior, with both covers shown to exist. This states the headline disproof. Only one direction is refuted; the dimension grows with $a,b$ . Permitted axioms: propext, Quot.sound, Classical.choice.

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The duality conjecture for metric entropy (Pietsch)

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Lean proof (OAI.MetricEntropyDuality.exists_entropy_duality_counterexample_with_covers)

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lean-proof link supplied by the VibeMathed registry.

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Artstein, Milman, Szarek, Tomczak-Jaegermann, On convexified packing and entropy duality (2004)

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Lean proof (OAI.MetricEntropyDuality.exists_entropy_duality_counterexample_with_covers)

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Artstein, Milman, Szarek, Tomczak-Jaegermann, On convexified packing and entropy duality (2004) evidence for this event

The duality conjecture for metric entropy (Pietsch) evidence for this event

Lean proof (OAI.MetricEntropyDuality.exists_entropy_duality_counterexample_with_covers) evidence for this event

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