The Hilbert basis of a convex cone C is a minimal set of integer vectors such that every integer vector in C is a conical combination of the vectors in the Hilbert basis with integer coefficients.
Given a lattice L ⊂ Z d and a convex polyhedral cone with generators a 1 , … , a n ∈ Z d
C = { λ 1 a 1 + … + λ n a n ∣ λ 1 , … , λ n ≥ 0 , λ 1 , … , λ n ∈ R } ⊂ R d we consider the monoid C ∩ L . By Gordan's lemma this monoid is finetely generated, i.e., there exists a finite set of lattice points { x 1 , … , x m } ⊂ C ∩ L such that every lattice point x ∈ C ∩ L is an integer conical combination of these points:
x = λ 1 x 1 + … + λ m x m , λ 1 , … , λ m ∈ Z , λ 1 , … , λ m ≥ 0 The cone C is called pointed, if x , − x ∈ C implies x = 0 . In this case there exists a unique minimal generating set of the monoid C ∩ L - the Hilbert basis of C. It is given by set of irreducible lattice points: An element x ∈ C ∩ L is called irreducible if it can not be written as the sum of two non-zero elements, i.e., x = y + z implies y = 0 or z = 0 .
