The face lattice of the polar of P

[done-with-fish]

Let $p be a full-dimensional Gorenstein polytope of dimension d and define

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$pd = polarize($p);

We know that the face lattices of $p and $pd are opposite to each other. My question is: given a face $f of $p, what face $pd does $f correspond to? Can polymake detect this face?

[blorenz]

What exactly do you mean by "detect this face" ?

The hasse diagram contains both an interior and an exterior description of any face and because of the homogenization in polymake it is very easy to construct the corresponding face and dual face.

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# the indices of the vertices defining face $i: $p->HASSE_DIAGRAM->FACES->[$i]; # the indices of the facets that are satisfied with equality at face $i: $p->HASSE_DIAGRAM->dual_faces->[$i]

Hence, you can define the face of $p as a new polytope via:

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$f = new Polytope(VERTICES=>$p->VERTICES->minor($p->HASSE_DIAGRAM->FACES->[$i],All));

And the corresponding dual face is defined by:

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$fd = new Polytope(VERTICES=>$p->FACETS->minor($p->HASSE_DIAGRAM->dual_faces->[$i],All));

If you first computed the HASSE_DIAGRAM (or at least VERTICES_IN_FACETS) of $p then the indices of vertices of $p correspond directly to indices of facets of the polar $pd (and vice versa).

Benjamin

[done-with-fish]

Thanks again for the help.

I was also curious if it is possible to construct the Gorenstein dual of a Gorenstein lattice polytope. While the polar of P is combinatorially opposite to P it is not itself a lattice polytope while the Gorenstein dual is combinatorially opposite and also a lattice polytope.