Minkowski inequality for convex hypersurfaces in Cartan–Hadamard manifolds

Prove the Minkowski inequality ∫_Γ H ≥ (n−1)|S^{n−1}|^{1/(n−1)}|Γ|^{(n−2)/(n−1)} for convex hypersurfaces in Cartan–Hadamard n-manifolds.

Background

For constant-mean-curvature hypersurfaces, the Minkowski inequality is equivalent to the sharp area estimate established earlier in the paper. The paper proves this equivalence and thereby verifies the inequality for embedded CMC hypersurfaces in dimension five.

The broader assertion for convex hypersurfaces is identified as a conjecture. The paper records that it has been established in dimension three, but does not resolve it in general dimensions.