Capacity inequalities and rigidity of cornered/conical manifolds

We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of P"olia-Szeg"o, Alexandrov-Fenchel and Penrose type inequalities in this setting. Among the techniques used in this paper are the inverse mean curvature flow for hypersurfaces with boundary.