Quasiconformal parametrization of metric surfaces with small dilatation

We verify a conjecture of Rajala: if (X, d) is a metric surface of locally finite Hausdorff 2-measure admitting some (geometrically) quasiconformal parametrization by a simply connected domain Ω ⊂ ℝ², then there exists a quasiconformal mapping f : X → Ω satisfying the modulus inequality 2π⁻¹ Mod Γ ≤ Mod fΓ ≤ 4π⁻¹ Mod Γ for all curve families Γ in X. This inequality is the best possible. Our proof is based on an inequality for the area of a planar convex body under a linear transformation which attains its Banach-Mazur distance to the Euclidean unit ball.