Constructing a polytope from a list of facets

[Nicolas]

Hello,

In the following example, I construct $p from a list of inequalities, then I define $q as the polytope with the same vertices as $p, and $r as the polytope with the same facets as $q. But $p->VERTICES and $r->VERTICES are different.

$p=new Polytope<Rational>;
$p->INEQUALITIES = <<".";
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
0 0 0 0 0
-1 1 0 0 1
1 -1 0 0 -1
-1 0 1 1 0
1 0 -1 -1 0
0 -1 0 -1 1
0 -1 -1 1 0
.
$q=new Polytope<Rational>(VERTICES=>$p->VERTICES);
$r=new Polytope<Rational>(FACETS=>$q->FACETS);
print $p->VERTICES;
print $r->VERTICES;

I'm new to polymake so I guess I missed something simple, but I can't see what it is.

Thanks for your help!
Nicolas

[herr]

Hi Nicolas,

The problem is that your polytope $p is not fulldimensional:

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polytope > print $p->DIM; 2

Since your polytope is 2-dimensional in R^4, its affine hull is nontrivial:

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polytope > print $p->AFFINE_HULL; -1 0 1 1 0 -1 1 0 0 1

Therefore, the representation of a facet is not unique. The polytope does not change if you add vectors of the affine hull to the facets. (Think of a line segment in 2D, which can be bounded by a lot of different lines.)
For example, we have

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print $r->FACETS->[0]+$p->AFFINE_HULL->[0]; 0 -1 -1 1 0

which is equal to $p->FACETS->[2].

In fact, your code is a little wrong. You are not allowed to define a polytope only by its FACETS (or VERTICES). If you want to define these properties, you always have to specify the AFFINE_HULL (or the LINEALITY_SPACE) as well (and you need to be sure that your description is irredundant). If you don't really know the polytope, you should define it via the properties INEQUALITIES or POINTS, see also the Tutorial on polytopes.

[Nicolas]

Got it! Thank you very much for your kind and detailed explanations.