Blaschke-Santaló type inequalities and quermassintegral inequalities in space forms

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-Santaló 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.