# Convex lattice polytope

 Four convex lattice polytopes in three dimensions Cube Cuboctahedron Octahedron Truncatedoctahedron (±1, ±1, ±1) (0, ±1, ±1) (0, 0, ±1) (0, ±1, ±2)

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 ZnRn. 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 ${\displaystyle A}$ which consists of all the exponent vectors appearing in a collection of monomials. For example, consider the polynomial of the form ${\displaystyle axy+bx^{2}+cy^{5}+d}$ with ${\displaystyle a,b,c,d\neq 0}$ has a lattice equal to the triangle
${\displaystyle {\rm {conv}}(\{(1,1),(2,0),(0,5),(0,0)\}).}$