Minkowski decomposition and generators of the moving cone for toric varieties

We prove that for smooth projective toric varieties, the Okounkov body of a T-invariant pseudo-effective divisor with respect to a T-invariant flag decomposes as a finite Minkowski sum of indecomposable polytopes, and that the set of these polytopes corresponds to a finite Minkowski basis whose elements span the extremal rays in the secondary fan. In fact, the Minkowski basis does not depend on the choice of the T-invariant flag. Moreover, we present an algorithm that computes the Minkowski basis.