  Let $\mathscr{L} \rightarrow X$ be an ample line bundle over a complex normal
projective variety $X$. We construct a flag $X_0 \subseteq X_1 \subseteq \cdots
\subseteq X_n=X$ of subvarieties for which the associated Okounkov body for
$\mathscr{L}$ is a rational polytope. In the case when $X$ is a homogeneous
surface, and the pseudoeffective cone of $X$ is rational polyhedral, we also
show that the global Okounkov body is a rational polyhedral cone if the flag of
subvarieties is suitably chosen. Finally, we provide an application to the
asymptotic study of group representations.
