Working with lattice polytopes

Given a polytope (Polyhedron) which is the convexHull of a given set of points we might ask if the polytope is a Cayley polytope

i1 : V = matrix {{0,2,-2,0},{0,1,1,1},{1,2,3,4}}

o1 = | 0 2 -2 0 |
     | 0 1 1  1 |
     | 1 2 3  4 |

              3        4
o1 : Matrix ZZ  <--- ZZ

i2 : P = convexHull V

o2 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 3
      number of facets => 4
      number of rays => 0
      number of vertices => 4

o2 : Polyhedron

i3 : isCayley P

o3 = true

We can also construct Cayley polytopes by taking the Cayley sum of several polytopes.

i4 : P2 = convexHull matrix{{0,1,2,3},{0,5,5,5},{1,2,3,2}}

o4 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 3
      number of facets => 4
      number of rays => 0
      number of vertices => 4

o4 : Polyhedron

i5 : cayley(P,P2,2)

o5 = {ambient dimension => 4           }
      dimension of lineality space => 0
      dimension of polyhedron => 4
      number of facets => 13
      number of rays => 0
      number of vertices => 8

o5 : Polyhedron

i6 : vertices oo

o6 = | 0 2 -2 0 0 1 3 2 |
     | 0 1 1  1 0 5 5 5 |
     | 1 2 3  4 1 2 2 3 |
     | 0 0 0  0 2 2 2 2 |

              4        8
o6 : Matrix QQ  <--- QQ