Talagrand’s operator-cotype bound
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Can a universal constant bound Rademacher operator cotype by the maximum of Gaussian cotype and the corresponding (q,1)-summing norm? The focused test is q=2.
Why it matters and what counts as progress
Why it matters: The comparison asks whether two established operator quantities control a third uniformly across spaces and dimensions.
What counts as progress: A universal comparison or a sequence of finite-dimensional operators whose ratio grows without bound, such as order √log n.
How the problem developed
- 17 Sept 2026 · Counterexample [1]
Xinglong Wu, reporting a counterexample discovered with ChatGPT GPT-5.6 Disproved Talagrand’s proposed universal operator-cotype inequality already at q=2; the paper gives finite-dimensional examples with a ratio growing at least as √log(n+1).
AI contribution
A Counterexample to Talagrand's Operator Cotype Problem
For sufficiently large n=2ᵏ, Wu’s Walsh–Hadamard norm and identity map Uₙ:Xₙ→ℓ∞ⁿ make C₂ʳ(Uₙ)/max(C₂ᵍ(Uₙ),‖Uₙ‖(2,1)) grow at least as c√log(n+1). Thus the proposed universal operator-cotype bound fails already at q=2.
Sources
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Publication dates and editorial changes.
ChatGPT-found example refutes Talagrand’s operator-cotype bound