In this talk I will discuss some recent work in progress with Alex Küronya. The aim is twofold. First of all I will give a local description of the theorem of Hu-Keel on the embedding of Mori Dream Spaces in toric varietis, extending it to the case of ample families.

After doing this, I will describe a method to prove finite generation of valuation semigroups for Newton-Okounkov bodies via tropical geometry, extening a result with Elisa Postinghel.