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We prove capacity inequalities involving the total mean curvature ofhypersurfaces with boundary in convex cones and the mass of asymptotically flatmanifolds with non-compact boundary. We then give the analogous ofP\"olia-Szeg\"o, Alexandrov-Fenchel and Penrose type inequalities in thissetting. Among the techniques used in this paper are the inverse mean curvatureflow for hypersurfaces with boundary.

Capacity inequalities and rigidity of cornered/conical manifolds