Description
A
statePolytope of an Ideal
I has as normalFan the Groebner fan of the ideal. We use the construction by Sturmfels, see Algorithm 3.2 in
Bernd Sturmfels' Groebner Bases and Convex Polytopes, volume 8 of University Lecture Series. American Mathematical Society, first edition, 1995.
Consider the following ideal in a ring with 3 variables:
i1 : R = QQ[a,b,c]
o1 = R
o1 : PolynomialRing

i2 : I = ideal (ab,ac,bc)
o2 = ideal (a  b, a  c, b  c)
o2 : Ideal of R

The state polytope of this ideal is a triangle in 3 space, because the ideal has three initial ideals:
i3 : statePolytope I
o3 = ({ b a ,  c a ,  c b }, {ambient dimension => 3 })
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 3
number of rays => 0
number of vertices => 3
o3 : Sequence

The generators of the three initial ideals are given in the first part of the result.