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I still do not understand what you are looking for, but maybe knowing that a polytope can be turned into a linear problem might help :

sage: D = polytopes.dodecahedron()
sage: p,x = D.to_linear_program(return_variable=True)
sage: p
Linear Program (no objective, 3 variables, 12 constraints)
sage: p.show()
Maximization:

Constraints:
  1/2 x2 + 1/4*sqrt5 + 1/4 x3 <= 1
  -1/2 x2 + 1/4*sqrt5 + 1/4 x3 <= 1
  1/2 x1 + 1/4*sqrt5 + 1/4 x2 <= 1
  1/2 x1 - 1/4*sqrt5 + 1/4 x2 <= 1
  1/4*sqrt5 + 1/4 x1 + 1/2 x3 <= 1
  1/4*sqrt5 + 1/4 x1 - 1/2 x3 <= 1
  -1/2 x1 + 1/4*sqrt5 + 1/4 x2 <= 1
  -1/2 x1 - 1/4*sqrt5 + 1/4 x2 <= 1
  -1/4*sqrt5 - 1/4 x1 + 1/2 x3 <= 1
  1/2 x2 - 1/4*sqrt5 + 1/4 x3 <= 1
  -1/2 x2 - 1/4*sqrt5 + 1/4 x3 <= 1
  -1/4*sqrt5 - 1/4 x1 - 1/2 x3 <= 1
Variables:
  x1 = x_0 is a continuous variable (min=-oo, max=+oo)
  x2 = x_1 is a continuous variable (min=-oo, max=+oo)
  x3 = x_2 is a continuous variable (min=-oo, max=+oo)