In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting points, can be written as the sum of exactly n lattice points in P. This property plays an important role in the theory of toric varieties, where it corresponds to projective normality of the toric variety determined by P.

Example

The simplex in R^k with the vertices at the origin and along the unit coordinate vectors is normal.

Relation to normal monoids

Any cancellative commutative monoid M can be embedded into an abelian group. More precisely, the canonical map from M into its Grothendieck group K(M) is an embedding. Define the normalization of M to be the set

where nx here means x added to itself n times. If M is equal to its normalization, then we say that M is a normal monoid. For example, the monoid Nn consisting of n-tuples of natural numbers is a normal monoid, with the Grothendieck group Zⁿ.

For a polytope P ⊆ R^k, lift P into R^k+1 so that it lies in the hyperplane x_k+1 = 1, and let C(P) be the set of all linear combinations with nonnegative coefficients of points in (P,1). Then C(P) is a convex cone,

If P is a convex lattice polytope, then it follows from Gordan's lemma that the intersection of C(P) with the lattice Z^k+1 is a finitely generated (commutative, cancellative) monoid. One can prove that P is a normal polytope if and only if this monoid is normal.

See also

* Convex cone

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

* Winfried Bruns, Joseph Gubeladze, Polytopes, rings and K-theory, preprint. Can be found at http://math.sfsu.edu/gubeladze/publications/kripo/kripo.pdf

Geometry