On the Willmore's Theorem for convex hypersurfaces

Let M be a compact convex hypersurface of class C2, which is assumed to bound a nonempty convex body K in the Euclidean space mathbf Rn and H be the mean curvature of M. We obtain a lower bound of the total square of mean curvature ∫M H2dA. The bound is the Minkowski quermassintegral of the convex body K. The total square of mean curvature attains the lower bound when M is an (n–1)-sphere.