  Building on work of Furstenberg and Tzkoni, we introduce ${\bf r}$-flag
affine quermassintegrals and their dual versions. These quantities generalize
affine and dual affine quermassintegrals as averages on flag manifolds (where
the Grassmannian can be considered as a special case). We establish affine and
linear invariance properties and extend fundamental results to this new
setting. In particular, we prove several affine isoperimetric inequalities from
convex geometry and their approximate reverse forms. We also introduce
functional forms of these quantities and establish corresponding inequalities.
