{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-27-banach-mazur-distance-cube-sharp-exponent-gpt-6/","md":"https://postcutoff.com/e/2026-10-27-banach-mazur-distance-cube-sharp-exponent-gpt-6/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-27-banach-mazur-distance-cube-sharp-exponent-gpt-6","date":"2026-10-27","date_precision":"day","short_title":"Sharp exponent 2/3 for the Banach–Mazur distance to the cube","deck":null,"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.36917: The sharp exponent for the Banach–Mazur distance to the cube (Friedland)","url":"https://arxiv.org/abs/2610.36917","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2608.21112: On the Banach–Mazur radius of ℓ∞ⁿ (Friedland, Naor, Youssef)","url":"https://arxiv.org/abs/2608.21112","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"arXiv 2609.27796: Gaussian polytopes with large Banach–Mazur distance to the cross-polytope (Hmadi)","url":"https://arxiv.org/abs/2609.27796","type":"paper","group":"primary","domain":"arxiv.org"}],"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Sorbonne Université","OpenAI"],"title":"The maximal Banach–Mazur distance to the cube is n^{2/3+o(1)}: Omer Friedland's sharp lower bound, with a key proposition proved by GPT-6","summary":"arXiv 2610.36917 (Omer Friedland, Institut de Mathématiques de Jussieu, 27 Oct 2026): \"We prove that R∞(n) ≥ cn^{2/3}(log(2n))^{−10}. Together with the upper bound of Friedland, Naor, and Youssef, this gives R∞(n) = n^{2/3+o(1)}\", where R∞(n) is the maximal Banach–Mazur distance from an n-dimensional real normed space to ℓ∞ⁿ. The extremal examples are polars of symmetric Gaussian polytopes with 2n generators (exponentially high probability). AI: the author formulated the variational framework and Proposition 6.1; \"OpenAI's ChatGPT (GPT-6) generated a proof of this proposition, which was used in preparing the proof presented here.\"","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)"],"key_numbers":[],"tags":["math","functional-analysis","asymptotic-geometric-analysis","banach-mazur-distance","convex-geometry","random-matrices","chatgpt","gpt-6"],"science":{"field":"mathematics","subfield":"functional analysis / asymptotic geometric analysis","problem":"Order of growth of the maximal Banach–Mazur distance R∞(n) from n-dimensional normed spaces to the cube ℓ∞ⁿ (Grünbaum's problem)","result":"R∞(n) ≥ c·n^{2/3}(log 2n)^{−10}; with the Friedland–Naor–Youssef upper bound, R∞(n) = n^{2/3+o(1)}","open_since":"1960","ai_system":["ChatGPT (GPT-6)"],"human_role":"Human-led: the author designed the strategy and stated the key proposition; GPT-6 generated its proof, which the author reworked","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed arXiv preprint"},"status":"pending","shock":""},"body_md":"## What happened\n\nHow 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\nupper bound is n. For decades the known bounds were about √n from below and n^{5/6} from above, until Tikhomirov reached\nn^{5/9} in 2019. In 2026 the gap closed quickly. Friedland raised the lower exponent to 4/7 and then 5/8 (Hmadi independently\nalso reached 5/8). In August Friedland, Naor and Youssef brought the upper bound down to n^{2/3}.\n\nThe new paper shows that this upper bound is sharp up to logarithms. Random Gaussian polytopes with 2n vertices, taken in polar\nform, 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\nformulated, was proved first by GPT-6 at his request.\n\n## Why it matters\n\nIt fixes the exponent of a classical quantity of Banach-space geometry, one that Bourgain, Szarek, Talagrand and Giannopoulos all\nworked on. It is also a clear case of the division of labour that maths papers now report: the human chooses the strategy and the\nstatement, and the model proves a lemma.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint","sources":[]}],"related":[{"id":"2026-10-29-ai-assisted-results-october-29-listing","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results in the 29 Oct 2026 sweep","deck":"Counterexamples to Wilking's 2007 question found by GPT-6 Astra, an LLM-written Plateau paper by De Lellis and Glaudo, explicit Reed–Solomon codes with optimal list size (Guruswami et al.), rank-four quiver mutation decided by GPT Astra, and a dozen named conjectures","takeaway":"The 29 Oct sweep covered the arXiv announcement of 28 Oct, which the previous run had read only through keyword queries. This time 43 papers with an AI statement surfaced, and a PDF scan of a further 405 new and cross-listed submissions in math, cs.CC, cs.DS and cs.DM found about 180 more with an AI statement, most of them describing more than editing help. Four results have their own entries: OpenAI's abundance claim and Jihao Liu's priority remark, OpenAI's Lean-checked L = RL = BPL, R(3,3,3,3) ≤ 61 and the sharp Banach–Mazur exponent. This page lists the rest. GPT-6 Astra found the counterexamples to Wilking's 2007 question 'with minimal human input' and discovered the main results of a rank-four quiver paper. Claude found examples in opinion dynamics and proposed a minimal-surface construction. An LLM supplied 'the main idea' of a paper by Camillo De Lellis, a member of the IAS advisory group on AI and mathematics, and wrote it.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","various researchers"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (science agent; found by a PDF scan of the 28 Oct arXiv announcement beyond keyword hits)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":null,"updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":180,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":119,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":210,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":149,"in_training_data":false}],"short_url":null}