Hemispheres minimize volume among small metric balls (optimal constant c_n)

Determine whether the optimal constant in Berger’s lower bound for the volume of small geodesic balls in n-dimensional Riemannian manifolds equals c_n = alpha_n/2; equivalently, prove that hemispheres minimize the Riemannian volume among small metric balls of a given dimension and intrinsic radius, so that for sufficiently small radius r one has vol(B_r(x)) ≥ (alpha_n/2) (2r/pi)^n for all points x in any n-dimensional Riemannian manifold.

Background

The paper recalls Berger’s result that there exists a uniform constant c_n such that for small metric balls B_r(x) in an n-dimensional Riemannian manifold, vol(B_r(x)) ≥ c_n (2r/pi)^n. The authors note a long-standing conjecture that identifies the optimal value of this constant as alpha_n/2, where alpha_n is the volume of the unit n-sphere.

They emphasize that the conjecture remains unresolved, even in dimension n = 2, and mention known partial results and related comparisons (e.g., cases where sectional curvature is bounded above by 1). This conjecture underpins the lower volume bounds used in parts of their analysis and motivates certain assumptions in their discretization and eigenvalue approximation results.