Anisotropic Alexandrov–Fenchel Type Inequalities and Hsiung–Minkowski Formula

In this paper, we introduce an anisotropic geometric quantity \(\mathbb {W}_{p,q;k} \) which involves the weighted integral of k-th elementary symmetric function. We first show the monotonicity of \({\mathbb {W}}_{p,1;k}\) and \({\mathbb {W}}_{0,q;k}\) along a class of inverse anisotropic curvature flows, and then prove the generalization of anisotropic Alexandrov–Fenchel type inequalities. On the other hand, an extension of anisotropic Hsiung–Minkowski formula is derived. Therefore, we at last obtain an extension of the Alexandrov–Fenchel type inequality, which involve the general \(\mathbb {W}_{p,q;k}\). In terms of the above inequalities, we have also demonstrated some other meaningful conclusions on convex body geometry, such as generalized \(L^p\)-Minkowski inequality and estimates of anisotropic p-affine surface area.