  We prove a generalization of Hsiung-Minkowski formulas for closed
submanifolds in semi-Riemannian manifolds with constant curvature. As a
corollary, we obtain volume and area upper bounds for k-convex hypersurfaces in
terms of a weighted total k-th mean curvature of the hypersurface. We also
obtain some Alexandrov-type results and eigenvalue estimates for hypersurfaces.
