Properties
- A lattice polytope is integrally closed if and only if it is normal and L is a direct summand of ℤd.
- A normal polytope can be made a into full-dimensional integrally closed polytope by changing the lattice of reference from ℤd to L and the ambient Euclidean space ℝd to the subspace ℝL.
- If a lattice polytope can be subdivided into normal polytopes then it is normal as well.
- If a lattice polytope in dimension d has lattice lengths greater than or equal to 4d(d+1) then the polytope is normal.
- If P is normal and φ:ℝd→ℝd is an affine map with φ(ℤd)=ℤd then φ(P) is normal.
- Proposition
P⊂ℝd a lattice polytope. Let C(P)=ℝ+(P,1)⊂ℝd+1 the following are equivalent:
- P is normal.
- The Hilbert basis of C(P)∩ℤd+1 =(P,1)∩ℤd+1
Conversely, for a full dimensional rational pointed cone C⊂ℝd if the Hilbert basis of C∩ℤd is in a hyperplane H⊂ℝd (dimH=d-1). Then C∩H is a normal polytope of dimension d-1.
Read more about this topic: Normal Polytope
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