  We compute Okounkov bodies of projective complexity-one T-varieties with
respect to two types of invariant flags. In particular, we show that the latter
are rational polytopes. Moreover, using results of Dave Anderson and Nathan
Ilten, we briefly exhibit explicit links to degenerations and T-deformations.
Finally, we prove that the global Okounkov body of a rational projective
complexity-one T-variety is rational polyhedral.
