The symmetric Mahler conjecture

Theorem 1.1: for every $n\ge1$ and every origin-symmetric convex body $K\subset\mathbb R^n$, $|K|\,|K^\circ|\ge4^n/n!$, with equality iff $K$ is an invertible linear image of a Hanner polytope. The proof uses a planar conformal lens (a rotation of Gross's uniform-distribution map), a holomorphic mass estimate proved by Stokes' theorem, and, for equality, a metric-median property of the norm and Hansen-Lima's classification. The companion Symplectic Balls in Symmetric Polar Products gives an independent proof of the inequality (not of the equality cases) through the Gromov width of $\mathrm{int}\,K\times\mathrm{int}\,K^\circ$. It does not treat the non-symmetric problem (a separate entry) and gives no stability estimate.

Theorem 1.1: for every $n\ge1$ and every origin-symmetric convex body $K\subset\mathbb R^n$, $|K|\,|K^\circ|\ge4^n/n!$, with equality iff $K$ is an invertible linear image of a Hanner polytope. The proof uses a planar conformal lens (a rotation of Gross's uniform-distribution map), a holomorphic mass estimate proved by Stokes' theorem, and, for equality, a metric-median property of the norm and Hansen-Lima's classification. The companion Symplectic Balls in Symmetric Polar Products gives an independent proof of the inequality (not of the equality cases) through the Gromov width of $\mathrm{int}\,K\times\mathrm{int}\,K^\circ$. It does not treat the non-symmetric problem (a separate entry) and gives no stability estimate.