  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. We prove that
these indecomposable polytopes form a Minkowski base and that they correspond
to the rays in the secondary fan. Moreover, we present an algorithm to find the
Minkowski base.
