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

Claims that every real Banach space with Markov type $p$ for at least one $p>1$ 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.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 23, 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: Naor's question: does nontrivial Markov type force an equivalent uniformly smooth norm (superreflexivity)? · Markov type and superreflexivity

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Unreleased internal OpenAI model
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VibeMathed record: Naor's question: does nontrivial Markov type force an equivalent uniformly smooth norm (superreflexivity)?

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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 $p$ for some $p>1$ , with the inequality required over all finite reversible chains and all times $n\ge1$ , 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.