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This paper continues the study of Alexandrov-Fenchel inequalities forquermassintegrals for $k$-convex domains. It focuses on the application to theMichael-Simon type inequalities for $k$-curvature operators. The proof usesoptimal transport maps as a tool to relate curvature quantities defined on theboundary of a domain.

Michael-Simon inequalities for $k$-th mean curvatures