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 \(\mathbb {H}^{n+1}\) ${\mathbb{H}}^{n+1}$ and \(\mathbb {S}^{n+1}\) ${\mathbb{S}}^{n+1}$ , which is a generalization of the quantitative Alexandrov–Fenchel inequality proved in \(\mathbb {R}^{n+1}\) ${\mathbb{R}}^{n+1}$ (VanBlargan and Wang, Commun Contemp Math 26(06):2350026, 2024). Then we use this method to derive the stability of some geometric inequalities involving weighted curvature integrals and quermassintegrals for nearly spherical sets in \(\mathbb {R}^{n+1}\) ${\mathbb{R}}^{n+1}$ and \(\mathbb {H}^{n+1}\) ${\mathbb{H}}^{n+1}$ .