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Vicente's question: is the Gromov width of every symmetric Lagrangian polar product equal to 4?

Theorem 1.1: for every $n\ge2$ and every origin-symmetric convex body $K\subset\mathbb R^n$ , $c_G(\mathrm{int}\,K\times\mathrm{int}\,K^\circ)=4$ , with explicit smooth symplectic embeddings of every ball of capacity $c<4$ ; no smoothness or strict convexity is assumed. The lower bound comes from a holomorphic ball principle (high-order vanishing gives large balls) and a uniform slice estimate for a conformal lens; the upper bound from a supporting-cylinder argument and nonsqueezing. Since symplectic embeddings preserve volume this reproves the symmetric Mahler inequality, but the paper makes no claim about equality cases. It says nothing about non-symmetric products or other capacities, and does not conflict with the Haim-Kislev-Ostrover counterexample to Viterbo's general conjecture.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 22, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Vicente's question: is the Gromov width of every symmetric Lagrangian polar product equal to 4? · Gromov width of symmetric polar products

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Vicente's question: is the Gromov width of every symmetric Lagrangian polar product equal to 4?

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No independent mathematician has checked this yet. Checked here: the abstract, introduction, background section and Theorem 1.1 of the TeX source, and Vicente's blog post itself, whose first listed question is exactly this one; the proof was not refereed. Lean-checked on the Comparator challenge SymmetricPolar (OAI.SymmetricPolar.symmetric_polar_main, OAI/Geometry/PolarProducts/Main.lean), listed in the release's formalization catalogue. Its statement, read here, says that for $n\ge2$ and every compact convex symmetric $K$ with nonempty interior, the Gromov width of $\mathrm{int}\,K\times\mathrm{int}\,K^\circ$ (supremum over smooth embeddings preserving the standard form $\omega_0$ of open balls $\pi(|q|^2+|p|^2)<c$ ) equals 4, and that every ball with $0<c<4$ embeds: the headline claim. An embedding at capacity exactly 4 is not asserted. Permitted axioms: propext, Quot.sound, Classical.choice. The statement was not independently audited and the development was not rebuilt here.