Newton-Okounkov bodies and Toric Degenerations of Mori dream spaces via   Tropical compactifications
  Given a smooth Mori dream space $X$ we construct a model dominating all the small $\mathbb{Q}$-factorial modifications via tropicalization. This construction allows us to recover a Minkowski basis for the Newton-Okounkov bodies of divisors on $X$ and hence the movable cone of $X$. The existence of such basis allows us to prove the polyhedrality of the global Newton-Okounkov body and the existence of toric degenerations. 
Elisa Postinghel and Stefano Urbinati
math.AG