  Perez proved some $L^2$ inequalities for closed convex hypersurfaces immersed
in the Euclidean space $\mathbb{R}^{n+1}$, more generally, for closed
hypersurfaces with non-negative Ricci curvature, immersed in an Einstein
manifold. In this paper, we discuss the rigidity of these inequalities when the
ambient manifold is $\mathbb{R}^{n+1}$, the hyperbolic space
$\mathbb{H}^{n+1}$, or the closed hemisphere $\mathbb{S}_+^{n+1}$. We also
obtain a generalization of the Perez's theorem to the hypersurfaces without the
hypothesis of non-negative Ricci curvature.
