In
algebraic geometry, a Newton–Okounkov body, also called an Okounkov body, is a
convex body in
Euclidean space associated to a
divisor (or more generally a linear system) on a
variety. The convex geometry of a Newton–Okounkov body encodes (asymptotic) information about the geometry of the variety and the divisor. It is a large generalization of the notion of the
Newton polytope of a projective
toric variety.
It was introduced (in passing) by
Andrei Okounkov in his papers in the late 1990s and early 2000s. Okounkov's construction relies on an earlier result of
Askold Khovanskii on semigroups of lattice points. Later, Okounkov's construction was generalized and systematically developed in the papers of
Robert Lazarsfeld and
Mircea Mustață as well as Kiumars Kaveh and Khovanskii.
Beside Newton polytopes of toric varieties, several polytopes appearing in representation theory (such as the
Gelfand–Zetlin polytopes and the string polytopes of Peter Littelmann and Arkady Berenstein–
Andrei Zelevinsky) can be realized as special cases of Newton–Okounkov bodies.