Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary. II

In this paper, we provide an affirmative answer to 16, Conjecture 1.5 on the Alexandrov-Fenchel inequality for quermassintegrals for convex capillary hypersurfaces in the Euclidean half-space. More generally, we establish a theory for capillary convex bodies in the half-space and prove a general Alexandrov-Fenchel inequality for mixed volumes of capillary convex bodies. The conjecture 16, Conjecture 1.5 follows as its consequence.