Sharp exponent 2/3 for the Banach–Mazur distance to the cube
Event confirmedAwaiting review
The takeaway
Omer Friedland (Sorbonne Université) proves that some n-dimensional normed spaces are at Banach–Mazur distance at least c·n^{2/3}/(log 2n)^10 from the cube ℓ∞ⁿ. With the n^{2/3} upper bound of Friedland, Naor and Youssef from August 2026, the maximal distance is n^{2/3+o(1)}. That settles the exponent in a problem going back to Grünbaum (1960), after a run of improvements from both sides: 5/9, 4/7 and 5/8 below, and 7/8, 5/6 and 2/3 above. The examples are polars of random Gaussian polytopes. Friedland says the strategy is his, and that ‘at the author’s request, OpenAI’s ChatGPT (GPT-6) generated a proof’ of a key proposition. Unrefereed preprint.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 29 October 2026
Your AI and this story
- GPT-6 Astra180 days after its cutoff
- Claude Opus 5.5119 days after its cutoff
- Gemini 4 Argon210 days after its cutoff
- Grok 4.7149 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 119 days before it.
Key facts
- Corollary 1.2: c·n^{2/3}/(log 2n)^{10} ≤ R∞(n) ≤ C·n^{2/3} for all n ≥ 2, so lim log R∞(n)/log n = 2/3; ‘The logarithmic power in the lower bound is not optimized’
- History (paper): the problem ‘was raised by Grünbaum’ (Trans. AMS 1960). Lower bounds: Szarek ≳ √n log n (1990), Tikhomirov exponent 5/9 (2019), Friedland 4/7 (arXiv 2602.10665), Friedland and independently Hmadi 5/8 (arXiv 2608.17743, 2609.27796). Upper bounds: trivial n, Bourgain–Szarek’s first asymptotic improvement (1988), Szarek–Talagrand 7/8, Giannopoulos 5/6, Friedland–Naor–Youssef n^{2/3} (arXiv 2608.21112, Aug 2026)
- Method: a joint minimization over coefficient representations and positive weights gives an exact second-variation identity and sparse coefficients; a Gaussian projection argument controls low-rank completions; an adaptive choice of exceptional generators reduces the rest to a Gaussian small-ball estimate over two finite nets
- AI statement (verbatim): ‘The research project, its main ideas, and the overall proof strategy are due to the author. The author formulated the variational framework and the statement of Proposition 6.1. At the author’s request, OpenAI’s ChatGPT (GPT-6) generated a proof of this proposition, which was used in preparing the proof presented here. ChatGPT was also used to explore simplifications, examine arguments, and assist with drafting and revising the exposition.’
- Proposition 6.1 fixes a ‘full record’ in Case (B) and bounds the probability of the corresponding record union by Ce^{−cr}; the final union bound uses it (Section 6)
- The companion papers in the 2026 sequence (2608.21112, 2608.17743, 2609.27796) carry no AI statement in their PDFs (scanned 29 Oct)
What happened
How far can an n-dimensional normed space be from the cube ℓ∞ⁿ, in the Banach–Mazur sense of the best linear fit? The trivial upper bound is n. For decades the known bounds were about √n from below and n^{5/6} from above, until Tikhomirov reached n^{5/9} in 2019. In 2026 the gap closed quickly. Friedland raised the lower exponent to 4/7 and then 5/8 (Hmadi independently also reached 5/8). In August Friedland, Naor and Youssef brought the upper bound down to n^{2/3}.
The new paper shows that this upper bound is sharp up to logarithms. Random Gaussian polytopes with 2n vertices, taken in polar form, are with overwhelming probability at distance at least n^{2/3} from the cube, up to logarithmic factors. One step in the proof, a small-ball estimate the author had formulated, was proved first by GPT-6 at his request.
Why it matters
It fixes the exponent of a classical quantity of Banach-space geometry, one that Bourgain, Szarek, Talagrand and Giannopoulos all worked on. It is also a clear case of the division of labour that maths papers now report: the human chooses the strategy and the statement, and the model proves a lemma.
What is disputed or not yet verified
| Verification | Unrefereed arXiv preprint |
|---|
Sources
3 sources from 1 site. Numbers match the chips in the text.
3 sources: 3 primary
Primary
- arXiv 2610.36917: The sharp exponent for the Banach–Mazur distance to the cube (Friedland)arxiv.org, paper
- arXiv 2608.21112: On the Banach–Mazur radius of ℓ∞ⁿ (Friedland, Naor, Youssef)arxiv.org, paper
- arXiv 2609.27796: Gaussian polytopes with large Banach–Mazur distance to the cross-polytope (Hmadi)arxiv.org, paper
Changes
- Filed (science agent; found by a PDF scan of the 28 Oct arXiv announcement beyond keyword hits)