Convex lattice polytope

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Polyhedron 6.png Polyhedron 6-8.png Polyhedron 8.png Polyhedron truncated 8.png
3D chess moves 111.png 3D chess moves 011.png 3D chess moves 001.png 3D chess moves 012.png
Cube Cuboctahedron Octahedron Truncated
(±1, ±1, ±1) (0, ±1, ±1) (0, 0, ±1) (0, ±1, ±2)
Four convex lattice polytopes in three dimensions

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.


  • 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 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[edit]


  • 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