On the uniqueness and continuity of the dual area measure

Abstract In this paper, the uniqueness of the dual Minkowski problem for the dual area measure is established via the dual Minkowski inequality and the dual log-Minkowski inequality. For real q > n − 1 , it is proved that the weak convergence of the q-th dual area measure implies the convergence of the corresponding convex bodies in the Hausdorff metric and that the solution to the dual Minkowski problem is continuous with respect to q.