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| A '''convex lattice polytope''' (also called '''Z-polyhedron''' or '''Z-polytope''') is a [[geometry|geometric]] object playing an important role in [[discrete geometry]] and [[combinatorial commutative algebra]]. It is a [[polytope]] in a Euclidean space '''R'''<sup>n</sup> which is a [[convex hull]] of finitely many points in the [[integer lattice]] '''Z'''<sup>n</sup> ⊂ '''R'''<sup>n</sup>. Such objects are prominently featured in the theory of [[toric variety|toric varieties]], where they correspond to polarized projective toric varieties.
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| == Examples == | |
| * An ''n''-dimensional [[simplex]] Δ in '''R'''<sup>n+1</sup> is the convex hull of ''n''+1 points that do not lie on a single affine hyperplane. The simplex is a convex lattice polytope if (and only if) the vertices have integral coordinates. The corresponding toric variety is the ''n''-dimensional [[projective space]] '''P'''<sup>n</sup>.
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| * The [[unit cube]] in '''R'''<sup>n</sup>, whose vertices are the ''2''<sup>n</sup> points all of whose coordinates are ''0'' or ''1'', is a convex lattice polytope. The corresponding toric variety is the [[Segre embedding]] of the ''n''-fold product of the projective line '''P'''<sup>1</sup>.
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| * In the special case of two-dimensional convex lattice polytopes in '''R'''<sup>2</sup>, they are also known as '''convex lattice polygons'''.
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| * In [[algebraic geometry]], an important instance of lattice polytopes called the '''Newton polytopes''' are the convex hulls of the set <math>A</math> which consists of all the exponent vectors appearing in a collection of monomials. For example, consider the polynomial of the form <math>axy+bx^2+cy^5+d</math> with <math>a,b,c,d \neq 0</math> has a lattice equal to the triangle
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| :<math>{\rm conv}(\{(1,1),(2,0),(0,5),(0,0)\}).\ </math>
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| == See also ==
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| * [[Normal polytope]]
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| * [[Pick's theorem]]
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| * [[Ehrhart polynomial]]
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| * [[Integer points in convex polyhedra]]
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| == References ==
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| * Ezra Miller, [[Bernd Sturmfels]], ''Combinatorial commutative algebra''. Graduate Texts in Mathematics, 227. Springer-Verlag, New York, 2005. xiv+417 pp. ISBN 0-387-22356-8
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| {{geometry-stub}}
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| [[Category:Polytopes]]
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| [[Category:Lattice points]]
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