Newton-Okounkov Bodies

The theory of Newton-Okounkov bodies attempts to generalize the correspondence between toric varieties and convex polytopes, to arbitrary varieties (even without presence of a group action). In this generality, one replaces convex polytopes, with convex bodies (i.e. convex compact subsets of Euclidean space). Beside Newton polytopes of toric varieties, many important examples of convex polytopes  e.g. moment polytopes (from symplectic geometry), Gelfand-Cetlin polytopes (and their generalization string polytopes) from representation theory fit into this general frame work.

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