Any smooth complex projective algebraic variety X can be made symplectic, by choosing an ample divisor D. A purely algebraic construction associates to D several convex polytopes, known as Okounkov bodies. I will report on work in progress, aimed at constructing a Liouville subdomain of X from each top-dimensional Okounkov body. The main feature is high control on the boundary Reeb dynamics, even if D is badly singular. Time permitting, I will mention potential applications to quantitative invariants and mirror symmetry.