  Atiyah's formulation of what is nowadays called the convexity theorem of
Atiyah-Guillemin-Sternberg has two parts: (a) the image of the moment map
arising from a Hamiltonian action of a torus on a symplectic manifold is a
convex polytope, and (b) all preimages of the moment map are connected. Part
(a) was generalized by Terng to the wider context of isoparametric submanifolds
in euclidean space. In this paper we prove a generalization of part (b) for a
certain class of isoparametric submanifolds (more precisely, for those with all
multiplicities strictly greater than 1). For generalized real flag manifolds,
which are an important class of isoparametric submanifolds, we give a
surjectivity criterium of a certain Kirwan map (involving equivariant
cohomology with rational coefficients) which arises naturally in this context.
Examples are also discussed.
