Browse › Convex and metric geometry › 087 The Mahler conjectures, functional inequalities and polar-product symplectic width
Symplectic Balls in Symmetric Polar Products
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Abstract
For every integer n ≥ 2 and every origin-symmetric convex body $K\subset\mathbb R^n$, we prove that the Gromov width of $\mathop{\mathrm{int}}\nolimits K\times\mathop{\mathrm{int}}\nolimits K^\circ$ is 4. We construct smooth symplectic embeddings of standard balls of every capacity $0\lt c\lt 4$ into this polar product. Volume preservation then resolves the symmetric Mahler conjecture positively in every dimension.
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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.
Comments and discussion
- Elliot Kienzle, Illustrating the AI proof of the Mahler conjecture (personal site, Oct 2026). Hand-drawn and interactive exposition of this paper's construction of large symplectic balls in $K\times K^\circ$, calling it "a clever construction applying a technique I haven't published yet"; Kienzle says he has not checked the details line by line. [family-level]
Formalization
- OpenAI math release, Lean scope note for family 087 (lean/docs/087.md). States that for $n\ge2$ the formalization gives Gromov width 4 for $\mathrm{int}\,K\times\mathrm{int}\,K^\circ$ and embeddings of every ball of capacity $0<c<4$, without asserting an embedding at capacity 4. [family-level]
- OpenAI math release, Main.lean (
symmetric_polar_main) (openai/math, commit adc7f12). Lean statement of Theorem 1.1 (width equal to 4 and embeddings below capacity 4); the comparator challenge isSymmetricPolar.lean.
Related prior literature mentioned by others
- Shiri Artstein-Avidan, Roman Karasev, Yaron Ostrover, From Symplectic Measurements to the Mahler Conjecture (arXiv:1303.4197; Duke Math. J. 163, 2014). Kienzle's exposition uses its Hofer–Zehnder capacity computation to explain the prior upper bound of 4, against which this paper supplies its ball-embedding lower bound.
Possibly relevant literature (found by the agent, not checked)
- Kun Shi, Guangcun Lu, The Viterbo's capacity conjectures for convex toric domains and the product of a 1-unconditional convex body and its polar (arXiv:2008.04000, 2020). Proves the Viterbo volume-capacity inequality for $K\times K^\circ$ with $K$ 1-unconditional, a special case near the paper's claimed width and volume-product consequence; not in its bibliography.
- Mark Berezovik, Roman Karasev, Symplectic polarity and Mahler's conjecture (arXiv:2211.14630, Israel J. Math. 2025). Gives a symplectic reformulation of Mahler's conjecture and capacity bounds for symplectically self-polar bodies, related to the paper's symplectic route to the volume-product bound; not in its bibliography.
Digests and expositions
- OpenAI Math Digest, Symplectic Balls in Symmetric Polar Products (Japanese exposition, undated). Illustrated reading guide tracking the holomorphic ball construction, lens estimate uniform near the tips, approximation and volume consequence, distinguishing capacities below 4 from an endpoint embedding. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 087
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026,
author = {{OpenAI}},
title = {{Symplectic Balls in Symmetric Polar Products}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026/paper.pdf}{OAI:Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/087.md. The Lean statements have not been checked by people against the paper's claims.
Family 087: The Mahler conjectures, functional inequalities and polar-product symplectic width
Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product $K\times K^\circ$ in dimension $2n$ has Gromov width 4.
Other papers in this family:
- The symmetric Mahler conjecture and its equality cases · September 22, 2026
- The Mahler Conjecture for General Convex Bodies · September 22, 2026