Okounkov bodies for ample line bundles

Let $\mathcal{L}\to X$ be an ample line bundle over a nonsingular complex projective variety $X$. We construct an admissable flag $X_0 \subseteq X_1 \subseteq…\subseteq X_n=X$ of subvarieties for which the associated Okounkov body for $\mathcal{L}$ is a rational polytope.