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.
released 2026-09-22 | 2 theorems · 8 lemmas · 14 proofs · 9,311 words |
PLAY LEVEL 1 »(pdf)
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.
released 2026-09-22 | 1 theorem · 26 lemmas · 32 proofs · 26,431 words |
PLAY LEVEL 2 »(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.
released 2026-09-22 | 2 theorems · 2 lemmas · 8 proofs · 6,815 words |
PLAY LEVEL 3 »(pdf)
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.