Stability of Alexandrov-Fenchel Type Inequalities for Nearly Spherical Sets in Space Forms

In this paper, we first derive a quantitative quermassintegral inequality for nearly spherical sets in ${\mathbb{H}}^{n+1}$ and ${\mathbb{S}}^{n+1}$, which is a generalization of the quantitative Alexandrov-Fenchel inequality proved in ${\mathbb{R}}^{n+1}$ [22]. Then we use this method to derive the stability of some geometric inequalities involving weighted curvature integrals and quermassintegrals for nearly spherical sets in ${\mathbb{R}}^{n+1}$ and ${\mathbb{H}}^{n+1}$.