Check, if inequality induces a facet

[markus.s]

Is there a built-in way/function to check, if some inequality induces a facet of a given polytope? (without printing all facets)

Right now, I print out the dimension of my given polytope (which I have as lp-file), then I add the inequality, which I want to check, as equality to the lp-file and print out the new dimension, thus I wonder if there is a more elegant solution possible.
(http://forum.polymake.org/viewtopic.php?f=8&t=317 seems to offer a way, but I am not to sure what to write instead of "INEQUALITIES=>cube(3)->FACETS/$H1" in my case)

Thanks!

Markus

[schroeter]

Is there a built-in way/function to check, if some inequality induces a facet of a given polytope?

No, there is no function like 'IS_FACET' in polymake.

If your polytope has full-dimension the best way is to check whether your inequality is in 'FACETS' or not. You can do this in a perl-script with an easy loop.

If you want to stick on your idea, first of all you must be sure that your inequality induces a face, that means solving a linear program in direction normal to your inequality with respect to your polytope $p.
The second step is to check the dimension of your face. You can construct a new polytope with an extra equation (in the example below 1 = 2*x + 3*y + 4*z) with

Code: Select all

$eq=new Vector(-1,2,3,4); $p_new=new Polytope(INEQUALITIES=>$p->FACETS,EQUATIONS=>$p->AFFINE_HULL/$eq);

and ask for the dimension.

[joswig]

If a given inequality is a facet of the feasible region can be decided by a linear program. This is dual to deciding whether one point from a set of points is a vertex of the convex hull. That latter problem is equivalent to deciding whether or not there is a hyperplane which separates that one point from all the other. This can be computed by the function separating_hyperplane.