{"id":"openai-math/Symplectic-Balls-in-Symmetric-Polar-Products","pdf":"https://github.com/openai/math/blob/main/preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026","tags":["symplectic-geometry","convex-geometry","gromov-width","mahler-conjecture"],"score":0,"title":"Symplectic Balls in Symmetric Polar Products","domain":"Convex and metric geometry","family":"087","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.MG","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-22","reactions":{},"formalized":true,"family_title":"The Mahler conjectures, functional inequalities and polar-product symplectic width"}