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$0$, 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.