Anisotropic capillary hypersurfaces in a wedge

We investigate anisotropic capillary hypersurfaces within a wedge in Euclidean space. In this study, we generalize the Minkowski norm $F$, traditionally employed to define the anisotropic surface energy, to a gauge on the unit sphere $S^n$. This generalization helps to illuminate a significant relationship between capillary hypersurfaces and hypersurfaces with free boundary. Our main results include new Minkowski formulae and a Heintze-Karcher type inequality. As an application, we prove an Alexandrov-type theorem, thereby extending the known results to the anisotropic setting.