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The duality conjecture for metric entropy (Pietsch)

For convex bodies A,BA,B let N(A,B)N(A,B) be the least number of translates of BB covering AA, and let K∘K^\circ be the polar of KK. The duality conjecture for metric entropy, originating in Pietsch's 1972 work on entropy numbers of operators and formulated geometrically by Artstein, Milman, Szarek and Tomczak-Jaegermann (2004, Conjecture 1), asks whether covering complexity is preserved under polarity up to universal changes of scale and constant factors. It was proved when one body is an ellipsoid (Artstein-Milman-Szarek 2004) and with logarithmic losses in general (E. Milman). Are there absolute constants a,b≥1a,b\ge1 such that 1blog⁡N(L∘,aK∘)≤log⁡N(K,L)≤blog⁡N(L∘,a−1K∘)\frac1b\log N(L^\circ,aK^\circ)\le\log N(K,L)\le b\log N(L^\circ,a^{-1}K^\circ) for all dimensions and all origin-symmetric convex bodies K,LK,L?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Asymptotic geometric analysis; covering numbers
Posed by
Albrecht Pietsch (1972, operator form); geometric form as Conjecture 1 of Artstein, Milman, Szarek and Tomczak-Jaegermann (GAFA 2004)
Year posed
1972
Years open
54y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
36 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every a,b≥1a,b\ge1 there exist nn and an origin-symmetric convex body K⊂RnK\subset\mathbb R^n such that, with LL the cube, log⁡N(K,L)>blog⁡N(L∘,a−1K∘)\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.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has a single manuscript.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture: for every a,b≥1a,b\ge1 there are nn and an origin-symmetric body K⊂RnK\subset\mathbb R^n with, for L=[−1,1]nL=[-1,1]^n, log⁡N(K,L)>blog⁡N(L∘,a−1K∘)\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 Rn\mathbb R^n, and states exactly Theorem 1.1 for a compact convex symmetric KK with nonempty interior, with both covers shown to exist. This states the headline disproof. Only one direction is refuted; the dimension grows with a,ba,b. Permitted axioms: propext, Quot.sound, Classical.choice.

Sources

Changelog1 change

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