Abstract. We establish a Schubert calculus for Bott-Samelson resolutions in the algebraic cobordism ring of a complete flag variety G/B extending the results of Bressler-Evens [4] to the algebro-geometric setting .

We compute the Newton-Okounkov bodies of line bundles on the complete flag variety of GL n for a geometric valuation coming from a flag of translated Schubert subvarieties. The Schubert subvarieties correspond to the terminal subwords in the decomposition (

We define convex-geometric counterparts of divided difference (or Demazure) operators from the Schubert calculus and representation theory. These operators are used to construct inductively polytopes that capture Demazure characters of representations of reductive groups. In particular, Gelfand-Zetlin polytopes and twisted cubes of Grossberg-Karshon are obtained in a uniform way.