  Suppose $E$ is fully symmetric Banach function space on $(0,1)$ or
$(0,\infty)$ or a fully symmetric Banach sequence space. We give necessary and
sufficient conditions on $f\in E$ so that its orbit $\Omega(f)$ is the closed
convex hull of its extreme points. We also give an application to symmetrically
normed ideals of compact operators on a Hilbert space.
