Weighted Alexandrov–Fenchel type inequalities for hypersurfaces in RnRⁿ

In this paper, we prove the following geometric inequalities in the Euclidean space , which are weighted Alexandrov–Fenchel type inequalities, provided that is a star‐shaped and ‐convex hypersurface. Equality holds if and only if is a coordinate sphere in . As an application, by letting in the above inequality, we obtain a lower bound for the outer radius in terms of the curvature integrals for star‐shaped and ‐convex hypersurfaces.