## Finding the cone-facet correspondence in the normal fan

Questions and problems about using polymake go here.
MattLarson
Posts: 7
Joined: 03 Jun 2018, 22:57

### Finding the cone-facet correspondence in the normal fan

If I have a (in my case unbounded) polytope, then I can compute the normal fan using normal_fan. Then there is an inclusion-reversing correspondence between the cones of the normal fan and the faces of the polytope.

Given a cone of the normal fan, I would like to find the corresponding face of the polytope. It seems like polymake must compute this bijection when finding the normal fan. Is there an easy/efficient way to find the corresponding face?

paffenholz
Developer
Posts: 186
Joined: 24 Dec 2010, 13:47

### Re: Finding the cone-facet correspondence in the normal fan

If computed with the function normal_fan from application fan, the rays of the normal fan come in the same order as the facets of the polytope, and the facet-vertex incidence matrix (FACETS_THRU_VERTICES) of the polytope has the same row and column order as the ray-maximal-cone incidence matrix MAXIMAL_CONES.

So if your cone is spanned by the rays with indices i_1, ..., i_k, then the corresponding face is the intersection of the facets with indices i_1, ..., i_k. The vertices spanning this face are the intersections of the rows with these indices (set intersection is done with a $*$ in polymake).

I don't think we have a function that tells you the index of the node in the HASSE_DIAGRAM of the polytope.

Andreas

MattLarson
Posts: 7
Joined: 03 Jun 2018, 22:57

### Re: Finding the cone-facet correspondence in the normal fan

Thank you so much!