  We observe some higher order Poincare-type inequalities on a closed manifold,
which is inspired by Hurwitz's proof of the Wirtinger's inequality using
Fourier theory. We then give some geometric implication of these inequalities
by applying them on the sphere. More specifically, by applying them to the
support function of a convex hypersurface in the Euclidean space, we obtain
some sharp Minkowski-type inequalities, such as a stability inequality for the
classical Minkowski inequality and the Alexandrov-Fenchel inequality.
