Klartag and Moshe prove the ε-Dvoretzky conjecture (polynomial dependence on ε); the key probabilistic idea came from a ChatGPT discussion
On Oct 2, 2026 Boaz Klartag and Shahar Moshe posted a short proof that the dependence on the accuracy ε in Dvoretzky's theorem is polynomial in 1/ε, not exponential. Their bound is n ≥ (C/ε)^((ℓ+1)/2)·|log ε| for an ℓ-dimensional section ε-close to an ellipsoid, through any interior point of any convex body. This settles a conjecture going back to V. Milman. The authors write that the idea of their new probability distribution on the Grassmannian "stemmed from a discussion between ChatGPT and the second named author".
Key facts
- arXiv 2610.03204 (math.FA), 'A polynomial bound in Dvoretzky's theorem', Oct 2, 2026
- Theorem 1.1: if n ≥ (C/ε)^((ℓ+1)/2)·|log ε|, every convex body K ⊂ R^n with 0 in its interior has an ℓ-dimensional section with F ⊆ K∩E ⊆ (1+ε)F for a centred ellipsoid F; Theorem 1.2 gets a Euclidean ball with the same bounds up to the constant
- Previous best (centrally symmetric case only): exp(Cℓ|log ε|/ε) (Paouris–Valettas, improving Schechtman and Gordon); all proofs since Milman's 1971 concentration-of-measure approach had exponential dependence on 1/ε already for ℓ = 3
- Example: a 3-dimensional section within ε = C·√(log n / n) of an ellipsoid, instead of C·log log n / log n
- Proof: probabilistic but simple, using a new random model on the Grassmannian, Brunn–Minkowski in the space of operators and a union bound; also a simultaneous version for finitely many bodies
- AI disclosure (acknowledgements): 'The idea to use the above probability distribution on the Grassmannian stemmed from a discussion between ChatGPT and the second named author. ChatGPT was additionally used for finding references, for inspecting and improving the proofs and for language editing.'
Science result
- Field
- mathematics / convex geometry / asymptotic geometric analysis
- Problem
- ε-Dvoretzky conjecture: polynomial dependence on 1/ε in Dvoretzky's theorem (V. Milman; Klartag–Novikov)
- Result
- Polynomial bound n ≥ (C/ε)^((ℓ+1)/2)|log ε| for nearly ellipsoidal ℓ-dimensional sections of arbitrary (not necessarily symmetric) convex bodies, and a Euclidean-ball version.
- AI system
- ChatGPT
- Human role
- Human-led. The key idea (the probability distribution on the Grassmannian) came out of a discussion between ChatGPT and Shahar Moshe; ChatGPT also checked and improved proofs.
- Verification
- Unrefereed preprint (Tel Aviv University / Weizmann Institute authors)
- Status
- pending
- Why surprising
- A central quantitative question about Dvoretzky's theorem fell to a short proof whose key idea the authors trace to a ChatGPT conversation.
What happened
Dvoretzky's theorem (around 1960, conjectured by Grothendieck) says that high-dimensional convex bodies have almost-Euclidean sections. For more than fifty years, every proof needed the ambient dimension to grow exponentially in 1/ε for a fixed section dimension. Klartag and Moshe give a short probabilistic proof with polynomial dependence that works for every convex body, symmetric or not. They name the source of the key idea, a new random model for choosing the subspace, in their acknowledgements: a conversation between ChatGPT and Moshe.
Why it matters
This is a classical problem in asymptotic geometric analysis, and the preprint comes from Boaz Klartag (Tel Aviv University and Weizmann Institute), a leading researcher in the field. The AI contribution is stated modestly, but it concerns the central idea rather than editing. As of Oct 5 we found no Hacker News or press coverage; the model version is not named.
Changelog
- 2026-10-05: created (06:30 run; arXiv paper missed by the sweep's date window, AI disclosure read in the PDF)
Related events
Sources (1)
id: 2026-10-02-epsilon-dvoretzky-conjecture-klartag-moshe · updated 2026-10-05 · open in the interactive timeline