Browse › Convex and metric geometry › 097 The Euclidean Steinitz–Bergström bound
The Euclidean Steinitz–Bergström theorem
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Abstract
Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝd has one signing for which every signed prefix has norm at most $C\sqrt d$, with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its unsigned partial sums. This determines the Euclidean Steinitz constant up to absolute factors, $S_2(d)=\Theta(\sqrt d)$, and resolves the Euclidean Steinitz–Bergström conjecture.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Matteo Russo and Jiaye Wei, The Online Steinitz Problem (arXiv:2610.16715, 12 October 2026). Develops online signing-to-ordering reductions and discrepancy bounds; its introduction notes that OpenAI announced a proof of the O(√d) Euclidean bound (citing "result 097"), and a footnote says the authors "have not independently verified the announced proof, which none of this paper's results requires."
Formalization
- OpenAI math release, Lean scope notes for 097: The Euclidean Steinitz–Bergström bound (lean/docs/097.md, 6 October 2026). The release's own Lean development, stating the prescribed-order signed prefix bound and the zero-sum reordering bound with one absolute constant C (including repeated and zero vectors), with comparator statement
lean/ComparatorChallenges/SteinitzBergstrom.lean. [AI-generated per source] - Erdős Problems / Plasma AI, Openai 2026 euclidean steinitz bergstrom theorem (formal-verification report, accessed 21 October 2026). Reports rebuilding
OAI.EuclideanSteinitzBergstrom.mainwith Lean 4.34.1, checking its axioms and comparator match for Theorems 1.1–1.2, and excluding the simplex lower bound and Section 6 corollaries from that verification. - anxkhn, Add the Euclidean Steinitz–Bergström prefix bounds (google-deepmind/formal-conjectures issue #6985, 8 October 2026). Proposes adding Theorems 1.1 and 1.2 of this paper as formal statements, with no
formal_proofattribute; the issue was still open when collected. - anxkhn, feat(Paper): add the Steinitz–Bergström bounds (google-deepmind/formal-conjectures PR #6990, opened 8 October 2026). Adds the statements of Theorems 1.1–1.2 only (the author says it was prepared with AI assistance); a reviewer approved on 10 October 2026, saying the shared constant and quantifier order match the manuscript, and the PR was still open when collected.
Related prior literature mentioned by others
- Nikhil Bansal and Aleksandar Nikolov, Discrepancy Theory: An Algorithmic and Geometric Perspective (arXiv:2608.00140, 31 July 2026). Russo and Wei cite this monograph as background on offline and online algorithmic discrepancy, context for the distinction between the paper's existential prefix-signing theorem and online guarantees.
Possibly relevant literature (found by the agent, not checked)
- Sankeerth Rao Karingula and Shachar Lovett, An elementary proof of the Komlós conjecture (arXiv:2609.20979, September 2026). Gives a simplified version of the Guo–Fang–Lu proof, whose directional-density and symmetrization method the paper adapts for its shifted-sign proposition; not in the paper's bibliography.
- Nikhil Bansal and Shashwat Garg, Algorithmic Discrepancy Beyond Partial Coloring (arXiv:1611.01805, 2016). Gives constructive bounds for the ℓ2 Steinitz problem, which bears on the paper's remark that its construction is existential and gives no efficient algorithm; not in the paper's bibliography.
Elsewhere
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 097
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026,
author = {{OpenAI}},
title = {{The Euclidean Steinitz--Bergstr{\"o}m theorem}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf}{OAI:The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/097.md. The Lean statements have not been checked by people against the paper's claims.
Family 097: The Euclidean Steinitz–Bergström bound
Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within $C\sqrt d$, independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order $S_2(d)=\Theta(\sqrt d)$.
This is the only paper in this family.