Convex lattice polytope - Alchetron, the free social encyclopedia

A convex lattice polytope (also called Z-polyhedron or Z-polytope) is a geometric object playing an important role in discrete geometry and combinatorial commutative algebra. It is a polytope in a Euclidean space Rⁿ which is a convex hull of finitely many points in the integer lattice Zⁿ ⊂ Rⁿ. Such objects are prominently featured in the theory of toric varieties, where they correspond to polarized projective toric varieties.

Examples

An n-dimensional simplex Δ in Rⁿ⁺¹ 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 Pⁿ.

The unit cube in Rⁿ, whose vertices are the 2ⁿ 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 P¹.

In the special case of two-dimensional convex lattice polytopes in R², they are also known as convex lattice polygons.

In algebraic geometry, an important instance of lattice polytopes called the Newton polytopes are the convex hulls of the set A which consists of all the exponent vectors appearing in a collection of monomials. For example, consider the polynomial of the form a x y + b x 2 + c y 5 + d with a , b , c , d ≠ 0 has a lattice equal to the triangle

c o n v ( { ( 1 , 1 ) , ( 2 , 0 ) , ( 0 , 5 ) , ( 0 , 0 ) } ) .

References

Convex lattice polytope Wikipedia

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