  We analyze marked poset polytopes and generalize a result due to Hibi and Li,
answering whether the marked chain polytope is unimodular equivalent to the
marked order polytope. Both polytopes appear naturally in the representation
theory of semi-simple Lie algebras, and hence we can give a necessary and
sufficient condition on the marked poset such that the associated toric
degenerations of the corresponding partial flag variety are isomorphic.
  We further show that the set of lattice points in such a marked poset
polytope is the Minkowski sum of sets of lattice points for 0-1 polytopes.
Moreover, we provide a decomposition of the marked poset into indecomposable
marked posets, which respects this Minkowski sum decomposition for the marked
chain polytopes polytopes.
