Newton–Okounkov body

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 far 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 Lazarsfeld and Mustata as well as Kaveh and Khovanskii.

Beside Newton polytopes of toric varieties, several polytopes appearing in representation theory (such as Gelfand–Zetlin polytopes and string polytopes of Littelmann and Berenstein–Zelevinsky) can be realized as special cases of Newton–Okounkov bodies.

References

Kaveh, Kiumars; Khovanskii, Askold (2012), "Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory", Annals of Mathematics, 176 (2): 925–978, doi:10.4007/annals.2012.176.2.5

Khovanskii, Askold (1992), "Newton polytope, Hilbert polynomial and sums of finite sets", Functional Analysis and Its Applications, 26: 276–281, doi:10.1007/bf01075048
Lazarsfeld, Robert; Mustata, Mircea (2008), "Convex bodies associated to linear series", Annales scientifiques de l'École normale supérieure, 42 (5): 783–835
Okounkov, Andrei (2003), Why would multiplicities be log-concave?, Progress in Math., 213, Boston, MA: Birkhauser
Okounkov, Andrei (1996), "Brunn-Minkowski inequality for multiplicities", Inventiones Mathematicae, 125 (3): 405–411, doi:10.1007/s002220050081

External links

Oberwolfach workshop "Okounkov bodies and applications"
BIRS workshop "Positivity of linear series and vector bundles"
BIRS workshop "Convex bodies and representation theory"
Oberwolfach workshop "New developments in Newton–Okounkov bodies"

Isaac Newton

Publications	De analysi per aequationes numero terminorum infinitas (1669, published 1711) Method of Fluxions (1671) De motu corporum in gyrum (1684) Philosophiæ Naturalis Principia Mathematica (1687) General Scholium (1713) Opticks (1704) The Queries (1704) Arithmetica Universalis (1707)

Other writings	Notes on the Jewish Temple Quaestiones quaedam philosophicae The Chronology of Ancient Kingdoms Amended (1728) An Historical Account of Two Notable Corruptions of Scripture (1754)

Newtonianism	Bucket argument Newton's inequalities Newton's law of cooling Newton's law of universal gravitation Post-Newtonian expansion Parameterized post-Newtonian formalism Newton–Cartan theory Schrödinger–Newton equation Gravitational constant Newton's laws of motion Newtonian dynamics Newton's method in optimization Gauss–Newton algorithm Truncated Newton method Newton's rings Newton's theorem about ovals Newton–Pepys problem Newtonian potential Newtonian fluid Classical mechanics Newtonian fluid Corpuscular theory of light Leibniz–Newton calculus controversy Rotating spheres Newton's cannonball Newton–Cotes formulas Newton's method Newton fractal Generalized Gauss–Newton method Newton's identities Newton polynomial Newton's theorem of revolving orbits Newton–Euler equations Newton number Kissing number problem Power number Solar mass Dynamics Absolute time and space Finite difference Table of Newtonian series Impact depth Structural coloration Inertia

Life	Cranbury Park Woolsthorpe Manor Early life of Isaac Newton Later life of Isaac Newton Religious views of Isaac Newton Isaac Newton's occult studies The Mysteryes of Nature and Art Scientific revolution Copernican Revolution

Friends and family	Catherine Barton John Conduitt William Clarke Benjamin Pulleyn William Stukeley William Jones Isaac Barrow Abraham de Moivre John Keill

Discoveries and inventions	Calculus Newton disc Newton polygon Newton–Okounkov body Newton's reflector Newtonian telescope Newton scale Newton's metal Newton's cradle Sextant

Phrases	"Hypotheses non fingo" "Standing on the shoulders of giants"

Theory expansions	Kepler's laws of planetary motion Problem of Apollonius

Related	Writing of Principia Mathematica Newton (Blake) List of things named after Isaac Newton Newton (unit) Isaac Newton in popular culture Elements of the Philosophy of Newton Isaac Newton S/O Philipose

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