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In this paper, we prove a generalized Minkowski inequality holds for anysmooth, $(k-1)$-convex, starshaped domain $\Omega.$ Our proof relies on thesolvability of the degenerate $k$-Hessian equation on the exterior domain$\mathbb R^n\setminus\Omega.$

Generalized Minkowski inequality via degenerate Hessian equations on exterior domains