In this paper, we first investigate weighted Minkowski type inequalities for nearly spherical sets in space forms, focusing on the sets that are $C^1$-close to geodesic spheres. Our results generalize the work of \cite{ G22} by incorporating broader geometric settings and convex weight functions. Additionally, we establish quantitative stability estimates for weighted Alexandrov-Fenchel type inequalities in $\mathbb{ R}^{ n+1}$ and $\mathbb{ H}^{ n+1}$, extending the earlier results of \cite{ VW24} and \cite{ ZZ23}. These inequalities hold for nearly spherical sets that are $W^{ 2,\infty}$-close to geodesic spheres coupled with general convex weights.
Categories: math.DG