Between two moments

In this short note, we draw attention to a relation between two Horn polytopes which is proved in [Chenciner-Jim\'enez P\'erez] as the result on the one side of a deep combinatorial result in [Fomin,Fulton, Li,Poon], on the other side of a simple computation involving complex structures. This suggested an inequality between Littlewood-Richardson coefficients which we prove using the symmetric characterization of these coefficients given in [Carr\'e,Leclerc].