Normal polytope

Normal polytope

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.

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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

\{ x \in K(M) \mid nx \in M,\ n\in\mathbb{N} \},

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,

C(P)=\{ \lambda_1(\textbf{x}_1, 1) + \cdots + \lambda_n(\textbf{x}_n, 1) \mid \textbf{x}_i \in P,\ \lambda_i \in \mathbb{R}, \lambda_i\geq 0\}.

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.

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References

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