Abstract:In his influential works, A. Okounkov showed how to associate a convex body to a very ample G-line bundle L on a projective G-variety X such that it projects to the moment polytope of X and the push-forward of the Lebesgue measure on it gives the Duistermaat-Heckamnn measure. He used this to prove the log-concavity of multiplicities in this case. Motivated by his work, recently Lazarsfeld-Mustata and Kaveh-Khovanskii developed a general theory of Newton-Okounkov bodies (without presence of a G-action). In the present talk, I will go back to the case where X has a G-action. After a brief review of the construction/results in the above works, I discuss how to associate different convex bodies to an arbitrary G-line bundle L on an arbitrary G-variety X which encode different information about X, L and the multiplicities of the G-action. Using this I will define the Duistermaat-Heckmann measure for L and prove a Brunn-Minkowski inequality for it. Also I will prove a Fujita approximation type result (from the theory of line bundles) for this Duistermaat-Heckmann measure. This talk is based on a preprint in preparation joint with A. G.