Vicente's question: is the Gromov width of every symmetric Lagrangian polar product equal to 4?

Theorem 1.1: for every $n\ge2$ and every origin-symmetric convex body $K\subset\mathbb R^n$, $c_G(\mathrm{int}\,K\times\mathrm{int}\,K^\circ)=4$, with explicit smooth symplectic embeddings of every ball of capacity $c<4$; no smoothness or strict convexity is assumed. The lower bound comes from a holomorphic ball principle (high-order vanishing gives large balls) and a uniform slice estimate for a conformal lens; the upper bound from a supporting-cylinder argument and nonsqueezing. Since symplectic embeddings preserve volume this reproves the symmetric Mahler inequality, but the paper makes no claim about equality cases. It says nothing about non-symmetric products or other capacities, and does not conflict with the Haim-Kislev-Ostrover counterexample to Viterbo's general conjecture.

Theorem 1.1: for every $n\ge2$ and every origin-symmetric convex body $K\subset\mathbb R^n$, $c_G(\mathrm{int}\,K\times\mathrm{int}\,K^\circ)=4$, with explicit smooth symplectic embeddings of every ball of capacity $c<4$; no smoothness or strict convexity is assumed. The lower bound comes from a holomorphic ball principle (high-order vanishing gives large balls) and a uniform slice estimate for a conformal lens; the upper bound from a supporting-cylinder argument and nonsqueezing. Since symplectic embeddings preserve volume this reproves the symmetric Mahler inequality, but the paper makes no claim about equality cases. It says nothing about non-symmetric products or other capacities, and does not conflict with the Haim-Kislev-Ostrover counterexample to Viterbo's general conjecture.