Generalized Minkowski inequality via degenerate Hessian equations on exterior domains

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