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Naor's question: does nontrivial Markov type force an equivalent uniformly smooth norm (superreflexivity)?

A normed space XX has Markov type pp with constant KK if for every finite stationary reversible Markov chain (Zt)(Z_t) on a set SS, every f:S→Xf:S\to X and every t≥1t\ge1, E∥f(Zt)−f(Z0)∥p≤Kp t E∥f(Z1)−f(Z0)∥p\mathbb E\|f(Z_t)-f(Z_0)\|^p\le K^p\,t\,\mathbb E\|f(Z_1)-f(Z_0)\|^p. Ball introduced Markov type for Lipschitz extension. Naor, Peres, Schramm and Sheffield showed that uniform smoothness of power type pp implies Markov type pp, so by Pisier's renorming every superreflexive space has Markov type pp for some p>1p>1. Markov type is a metric property, so a converse would give a metric characterization of superreflexivity in the Ribe program. Naor (2012, Question 4) asked: can a Banach space have nontrivial Markov type (Markov type pp for some p>1p>1) without admitting an equivalent uniformly smooth norm?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Banach space geometry; Ribe program; metric invariants
Posed by
Assaf Naor, An introduction to the Ribe program, Japanese Journal of Mathematics 7 (2012), Question 4
Year posed
2012
Years open
14y
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
28 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims that every real Banach space with Markov type pp for at least one p>1p>1 admits an equivalent uniformly convex norm, hence is superreflexive and (by Pisier) admits an equivalent uniformly smooth norm. With the known converse (Naor-Peres-Schramm-Sheffield plus Pisier) this characterizes superreflexivity by nontrivial Markov type, answering Naor's question negatively. The proof builds, from a nonreflexive space finitely representable in a non-superreflexive one, a spreading norm and explicit reversible walks whose image has linear displacement. It does NOT address complex scalars separately, the exact Markov type exponent of a given space, or Ball's Markov type 2 questions for specific spaces.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. The main theorem has a Lean formalization listed in the release's formalization catalogue.

Verification

No independent mathematician has checked this yet. The main theorem was read against Naor's Question 4. formalization.yaml lists ComparatorChallenges/MarkovType.json with declaration OAI.MarkovSuperreflexivity.hasNontrivialMarkovType_iff_hasEquivalentUCNorm in OAI/Analysis/MarkovType/Main.lean. The comparator statement, read here, says that for every real Banach space (CompleteSpace), having Markov type pp for some p>1p>1, with the inequality required over all finite reversible chains and all times n≥1n\ge1, is equivalent to admitting an equivalent uniformly convex norm. Given Pisier's duality between uniformly convex and uniformly smooth renormings, which the formal statement does not itself include, this is the headline. Not rebuilt here.

Sources

Changelog1 change

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