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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. Contents 1 Example 2 Relation to normal monoids 3 See also 4 References Example The simplex in Rk 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  ⊆ Rk, lift P into Rk+1 so that it lies in the hyperplane xk+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 Zk+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 This geometry-related article is a stub. You can help Wikipedia by expanding it.v · d · e