Extracting inequalities for polytopes

edit

Comme ça :

sage: P = polytopes.simplex(4)                                                    
sage: P.inequalities_list()                                                     
[[1, 0, -1, -1, -1, -1],
 [0, 0, 1, 0, 0, 0],
 [0, 0, 0, 1, 0, 0],
 [0, 0, 0, 0, 1, 0],
 [0, 0, 0, 0, 0, 1]]

The documentation for Polyhedron can be accessed using ?:

sage: Polyhedron?

or online:

• SageMath documentation: polyhedra

See some examples at

• Ask Sage question 53419: Triple integrals in a specific region of space

Let us look at the specific polytope in the question.

The H-representation consists in

• equalities of the form A * x + b = 0
• inequalities of the form A * x + b >= 0

To get the vector A and the scalar b, use the A and b methods of equalities and inequalities.

sage: eq0 = rep[0]
sage: eq0

sage: eq0.A()
(0, -3, -3/2*sqrt5 - 3/2)
sage: eq0.b()
3/4*sqrt5 + 11/4

sage: eq10 = rep[10]
sage: eq10
An inequality (-132*sqrt5 - 300, 0, -84*sqrt5 - 180) x + 122*sqrt5 + 270 >= 0

sage: eq10.A()
(-132*sqrt5 - 300, 0, -84*sqrt5 - 180)
sage: eq0.b()
3/4*sqrt5 + 11/4

If you need radical expressions (when they exist), change to the symbolic ring:

sage: A10, b10 = eq10.A(), eq10.b()
sage: A10.change_ring(SR), SR(b10)
((-132*sqrt(5) - 300, 0, -84*sqrt(5) - 180), 122*sqrt(5) + 270)