Schubert calculus on Newton–Okounkov polytopes

Theory of Newton–Okounkov convex bodies can be used to extend toric geometry to varieties with more general reductive group actions. In particular, any Newton–Okounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent Schubert cycles by linear combinations of faces of the polytopes so that the intersection product of cycles corresponds to the set-theoretic intersection of faces (whenever the latter are transverse). I will talk about particular realizations of this approach in types $A$, $B$ and $C$.