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The cotype-cotype conjecture: finite cotype of a Banach space and its dual implies K-convexity (approximation property case)

A Banach space XX is KK-convex if the Rademacher projections on L2({−1,1}s;X)L_2(\{-1,1\}^s;X) are uniformly bounded; by Maurey-Pisier and Pisier this is equivalent to nontrivial Rademacher type. If XX is KK-convex then XX and X∗X^* have finite cotype. Pisier (1980-81) proved the converse when XX has the bounded approximation property and the cotype exponents q,rq,r of X,X∗X,X^* satisfy a restriction, and asked whether the restriction can be removed (Remark 2(iii)). Without any approximation hypothesis the converse fails (Pisier 1983). Szarek and Tomczak-Jaegermann (2009) call this the cotype-cotype conjecture, and Gupta, Misra and Ray (2025, Conjecture 1.5) state the BAP form. If XX has the (bounded) approximation property and both XX and X∗X^* have finite cotype, with arbitrary exponents, must XX be KK-convex?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Banach space geometry; Rademacher type and cotype, K-convexity
Posed by
G. Pisier, Seminaire d'analyse fonctionnelle 1980-81, expose VII, Remark 2(iii); named the cotype-cotype conjecture by Szarek and Tomczak-Jaegermann (2009)
Year posed
1981
Years open
45y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
26 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: a nonzero real Banach space XX with the approximation property is KK-convex if and only if Cq(X)<∞C_q(X)<\infty and Cr(X∗)<∞C_r(X^*)<\infty for some q,r∈[2,∞)q,r\in[2,\infty), possibly different; equivalently such XX has nontrivial Rademacher type. Corollary 1.2 makes the bound on K(X)K(X) uniform in the cotype data and gives dimension-free metric-entropy duality for the corresponding convex bodies; a further consequence settles a Gupta-Misra-Ray tensor-norm conjecture (their Conjecture 1.4) under AP. Real scalars only; constants are existential. It does not address spaces failing AP, where the statement is false.

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.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Pisier's question and the BAP formulation of Gupta-Misra-Ray. The result assumes only the ordinary approximation property, which is weaker than BAP, so the BAP conjecture is covered; the conjecture without an approximation hypothesis is false (Pisier 1983), as the paper says. The family is in lean/formalization.yaml (ComparatorChallenges/Cotype.json, declaration OAI.Cotype.mainTarget_proved, file OAI/Analysis/Cotype/Main.lean). The comparator statement was read here: for every nontrivial real Banach space with the approximation property, K-convexity (uniform bound on finite-cube Rademacher projections) is equivalent to finite cotype of XX and of its continuous dual. This is the headline claim. Not rebuilt here.

Sources

Changelog1 change

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