Note on cohomology rings of spherical varieties and volume polynomial

Let G be a complex reductive group and X a projective spherical G-variety. Moreover, assume that the subalgebra A of the cohomology ring H^*(X, R) generated by the Chern classes of line bundles has Poincare duality. We give a description of the subalgebra A in terms of the volume of polytopes. This generalizes the Khovanskii-Pukhlikov description of the cohomology ring of a smooth toric variety. In particular, we obtain a unified description for the cohomology rings of complete flag varieties and smooth toric varieties. As another example we get a description of the cohomology ring of the variety of complete conics. We also address the question of additivity of the moment and string polytopes and prove the additivity of the moment polytope for complete symmetric varieties.