Anisotropic weighted isoperimetric inequalities for star-shaped and $F$-mean convex hypersurface

We prove two anisotropic type weighted geometric inequalities that hold for star-shaped and $F$-mean convex hypersurfaces in ${\mathbb{R}}^{n+1}$. These inequalities involve the anisotropic $p$-momentum, the anisotropic perimeter and the volume of the region enclosed by the hypersurface. We show that the Wulff shape of $F$ is the unique minimizer of the corresponding functionals among all star-shaped and $F$-mean convex sets. We also consider their quantitative versions characterized by the Hausdorff distance between the hypersurface and a rescaled Wulff shape. As a corollary, we obtain the stability of the Weinstock inequality for star-shaped and strictly mean convex domains, which requires weaker convexity compared to \cite{Gavitone}.