Generalized Minkowski formulas and rigidity results for anisotropic capillary hypersurfaces

In this paper, we obtain a new Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces in the half-space, which includes the weighted Hsiung-Minkowski formula and classical anisotropic Minkowski identity for closed hypersurfaces as special cases. As applications, we prove some anisotropic Alexandrov-type theorems and rigidity results for anisotropic capillary hypersurfaces. Specially, the uniqueness of the solution to the anisotropic Orlicz-Christoffel-Minkowski problem is obtained, and thus a new proof is provided for the uniqueness of the solution to $L_p$-Minkowski problem with $p\geq 1$ in the Euclidean capillary convex bodies geometry.