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. (Lean)
Manuscripts (3)
The symmetric Mahler conjecture and its equality cases
September 22, 2026 · source · Lean formalization available · cite as \cite{OAI:The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026}
We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in ℝn has volume product at least 4^n/n!, with equality exactly for invertible linear images of Hanner polytopes.
The Mahler Conjecture for General Convex Bodies
September 22, 2026 · source · cite as \cite{OAI:The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026}
We resolve the Mahler conjecture for general convex bodies positively. For every convex body K\subset\mathbb R^n, n ≥ 1, with Santaló point s(K), |K|\,|(K-s(K))^\circ|\ge (n+1)^{n+1}/(n!)^2, with equality exactly for simplices.
Symplectic Balls in Symmetric Polar Products
September 22, 2026 · source · Lean formalization available · cite as \cite{OAI:Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026}
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.
Result 087 · Convex and metric geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md