Description
This function computes the regular cell decomposition of
P given by the weight vector
w. This is computed by placing the ith lattice point of
P on height
w_i in n+1 space, taking the convexHull of these with the ray (0,...,0,1), and projecting the compact faces into n space. Note that the polyhedron must be compact, i.e. a polytope and the length of the weight vector must be the number of lattice points.
i1 : P = crossPolytope 3
o1 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 8
number of rays => 0
number of vertices => 6
o1 : Polyhedron

i2 : w = matrix {{1,2,2,2,2,2,1}}
o2 =  1 2 2 2 2 2 1 
1 7
o2 : Matrix ZZ < ZZ

i3 : L = cellDecompose(P,w)
o3 = {{ambient dimension => 3 }, {ambient dimension => 3
dimension of lineality space => 0 dimension of lineality space =>
dimension of polyhedron => 3 dimension of polyhedron => 3
number of facets => 4 number of facets => 4
number of rays => 0 number of rays => 0
number of vertices => 4 number of vertices => 4

}, {ambient dimension => 3 }, {ambient dimension => 3
0 dimension of lineality space => 0 dimension of lineality space
dimension of polyhedron => 3 dimension of polyhedron => 3
number of facets => 4 number of facets => 4
number of rays => 0 number of rays => 0
number of vertices => 4 number of vertices => 4

}}
=> 0
o3 : List

i4 : apply(L,vertices)
o4 = { 1 1 0 0 ,  1 1 0 0 ,  1 1 0 0 ,  1 1 0 0 }
 0 0 1 0   0 0 1 0   0 0 1 0   0 0 1 0 
 0 0 0 1   0 0 0 1   0 0 0 1   0 0 0 1 
o4 : List
