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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.

Manuscripts in this family · 3

The symmetric Mahler conjecture and its equality cases

Sep 22, 2026·paper.pdfLean

We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in ℝ

^{n}

has volume product at least

4^n/n!

, with equality exactly for invertible linear images of Hanner polytopes.

Full abstract

PDF on GitHub LaTeX source

The Mahler Conjecture for General Convex Bodies

Sep 22, 2026·paper.pdf

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.

Full abstract

PDF on GitHub LaTeX source

Symplectic Balls in Symmetric Polar Products

Sep 22, 2026·paper.pdfLean

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.

Full abstract

PDF on GitHub LaTeX source

OpenAI also released the model's reasoning trace for this result: reasoning trace (PDF).