Optimal constant in Berger’s antipodal displacement inequality on S^2

Determine the exact optimal constant C such that, for every Riemannian metric g on the 2‑sphere S^2 with distance function d_g and every fixed‑point‑free involution α: S^2 → S^2, the inequality Area(S^2,g) ≥ C [min_{x∈S^2} d_g(x,α(x))]^2 holds. Establish whether the optimal constant equals 4/π, as suggested by the antipodal map on the round sphere.

Background

Berger proved that for any Riemannian metric on the 2‑sphere and any fixed‑point‑free involution, the area admits a curvature‑free lower bound in terms of the involution’s minimal displacement, with some positive constant ≥ 1/2. The present paper extends related bounds to convex hypersurfaces in all dimensions and proves a sharp mean‑width inequality.

The precise best constant in Berger’s 2‑dimensional inequality remains undetermined; geometric evidence points to the value 4/π given by the round sphere with the antipodal involution. Determining this constant would settle the sharp form of this classical isoembolic‑type bound in dimension two and would interact with other sharp inequalities (e.g., via Minkowski’s inequality).