# Convex lattice polytope

Jump to navigation
Jump to search

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 **R**^{n} which is a convex hull of finitely many points in the integer lattice **Z**^{n} ⊂ **R**^{n}. 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**R**^{n+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**P**^{n}. - The unit cube in
**R**^{n}, whose vertices are the*2*^{n}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**^{1}. - In the special case of two-dimensional convex lattice polytopes in
**R**^{2}, 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 which consists of all the exponent vectors appearing in a collection of monomials. For example, consider the polynomial of the form with has a lattice equal to the triangle

## See also

## References

- 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