A convex lattice 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 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.

See also

* Normal polytope
* Pick's theorem
* Ehrhart polynomial

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