Weight distribution of Iwasawa projection

In the setting of a connected noncompact semisimple Lie group with a finite center, we prove a property of the Iwasawa decomposition, and use it to show that, in Kostant's convexity result [7], the Iwasawa projection puts almost all its weight near a single vertex of the convex hull. As another application, we will revisit the results of Huang-Tam [10], [5] on the iteration of group elements under the Cartan and the Iwasawa decompositions.