# How to find the facet inequalities for Bell-Wigner polytope?

The Bell-Wigner polytope has the following extreme points

$(1,0,0,0,0,0)$
$(0,1,0,0,0,0)$
$(0,0,1,0,0,0)$
$(1,1,0,1,0,0)$
$(1,0,1,0,1,0)$
$(0,1,1,0,0,1)$
$(1,1,1,1,1,1)$
$(0,0,0,0,0,0)$

I checked the largest set of independent position vectors consists of 7 points thus it's a six dimensional polytope. I wanted to find all the facet inequalities. I was thinking of the following method

For each selection of six points out of the given eight points do the following : see if they are linearly independent, if they are then obtain five independent vectors from them and form the 5 dimensional hyperplane they define and check if all the extreme eight points lie on one side of this hyperplane, if yes it defines a facet.

Am I correct ? Also can any kind of symmetry or some other thing help me simplify my method ?

Yes that will work, if you change the word "extreme eight points" to "all 8 points" in your pseudo-code, and since you only have 8 points in 6 dimensional space, the number of facets will be very small and at any rate, you only have ${8 \choose 2} = 28$ subsets of points to check for facet inequalities. If you ever have a situation where you want the inequality description for a convex hull/polytope where the number of points minus the dimension is considerably higher than just 2, update your question and I can give you some sophisticated methods (that use computers, of course). However your case is so trivial with 8 points in 6 dimensions, brute force is the easiest way to go, especially with a computer.