  In this paper, we prove a family of identities for closed and strictly convex
hypersurfaces in the sphere and hyperbolic/de Sitter space. As applications, we
prove Blaschke-Santal\'o type inequalities in the sphere and hyperbolic/de
Sitter space, which generalizes the previous work of Gao, Hug and Schneider
\cite{GHS03}. We also prove the quermassintegral inequalities in hyperbolic/de
Sitter space.
