摘要：

       In this talk, we study the Brunn-Minkowski inequality for the $q$-th dual quermassintegral with $q>n$, a problem recently posed by Sadovsky and Zhang. We show that the inequality fails for arbitrary convex bodies in the full range $q>n$, and for origin-symmetric convex bodies when $q>n+2$. At the endpoint $q=n+2$, we prove the inequality for origin-symmetric convex bodies by applying Hadwiger's inequality for the polar moment of inertia. Moreover, for unconditional convex bodies, we establish the inequality for the full range $0<q\le n+1$, using a singular weighted Reilly formula and coordinate-slice Hardy inequalities. As applications, we derive uniqueness results for the associated dual curvature measures. This talk is based on joint work with Haizhong Li and Yijia Zhang.