  On the polytope defined in Feigin, Fourier, and Littelmann (2011), associated
to any rectangle highest weight, we define a structure of an type
$A_n$-crystal. We show, by using the Stembridge axioms, that this crystal is
isomorphic to the one obtained from Kashiwara's crystal bases theory. Further
we define on this polytope a bijective map and show that this map satisfies the
properties of a weak promotion operator. This implies in particular that we
provide an explicit realization of Kirillov-Reshetikhin crystals for the affine
type $A^{(1)}_n$ via polytopes.
