The Mahler conjectures, functional inequalities and polar-product symplectic width

What OpenAI claims

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× K° in dimension 2n has Gromov width 4. (Lean)

Papers (3)

1. The symmetric Mahler conjecture and its equality cases

   22 September 2026 · Preprint, not peer reviewed · PDF

   We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in ℝn has volume product at least 4ⁿ/n!, with equality exactly for invertible linear images of Hanner polytopes.

2. The Mahler Conjecture for General Convex Bodies

   22 September 2026 · Preprint, not peer reviewed · PDF

   We resolve the Mahler conjecture for general convex bodies positively. For every convex body K ⊂ ℝⁿ, n ≥ 1, with Santaló point s(K), |K| |(K-s(K))°| ≥ (n+1)ⁿ⁺¹/(n!)², with equality exactly for simplices.

3. Symplectic Balls in Symmetric Polar Products

   22 September 2026 · Preprint, not peer reviewed · PDF

   For every integer n ≥ 2 and every origin-symmetric convex body K ⊂ ℝⁿ, we prove that the Gromov width of int K×int K° is 4. We construct smooth symplectic embeddings of standard balls of every capacity 0 < c < 4 into this polar product. Volume preservation then resolves the symmetric Mahler conjecture positively in every dimension.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.