Lusztig polytopes and FFLV polytopes

In this paper we prove that in type $\tt A_n$, the Feigin-Fourier-Littelmann-Vinberg (FFLV) polytope coincides with the Minkowski sum of Lusztig polytopes arising from various reduced decompositions. Using this result, we formulate a conjecture about the crystal structures on FFLV polytopes.
Comment: 22 pages, 12 figures; v2: added Proposition 5.2 and 5.4