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### Strugles with VERTICES_IN_FACETS

Posted: 06 Feb 2019, 17:34
Hello,
I am a new user on polymake and need some help.
Here is my problem:
My purpose is to compute a cone \$c pointed at some vertex \$v of a polytope \$p, such that its rays are the affine hull of the facets of \$p incident to \$v, and such that \$p itself is not contained in \$c. So my question is: how can I keep track of the facets that are incident to my vertex \$v, meaning to get their inequalities and affine hull. VERTICES_IN_FACETS and FACETS_THRU_VERTICES only seems to show the labels of the vertices and facets not information on them. Or am I wrong?

Thank you very much, and apologies for my english.

Rado

### Re: Strugles with VERTICES_IN_FACETS

Posted: 07 Feb 2019, 13:59
My purpose is to compute a cone \$c pointed at some vertex \$v of a polytope \$p, such that its rays are the affine hull of the facets of \$p incident to \$v, and such that \$p itself is not contained in \$c.
Sorry, I do not understand what you want to compute. Do you want to compute the intersection of the affine halfspaces given by the facets through a fixed vertex? This would be like the following.

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```> \$c=cube(3); > print \$c->VERTICES->; 1 -1 -1 -1 ```
Just to check which vertex is indexed with 0. Here you see the indices of the facets containing that vertices (as row indices to the FACETS matrix).

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```> print \$c->FACETS_THRU_VERTICES->; {0 2 4} ```
If you want to get the inequalities:

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```> print \$c->FACETS->minor(\$c->FACETS_THRU_VERTICES->,All); 1 1 0 0 1 0 1 0 1 0 0 1 ```
And you can produce a new object of type Polytope, of course:

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```\$C=new Polytope(FACETS=>\$c->FACETS->minor(\$c->FACETS_THRU_VERTICES->,All)); ```
Here using FACETS (instead of the standard INEQUALITIES) as an input property is OK, because you know that there are no hidden redundancies.

### Re: Strugles with VERTICES_IN_FACETS

Posted: 07 Feb 2019, 15:57
Thank you for your fast reply,
Sorry, I do not understand what you want to compute. Do you want to compute the intersection of the affine halfspaces given by the facets through a fixed vertex?
This is indeed what I aim for but I want to consider the closed halfspaces that does not contain the polytope. That is, for a fixed vertex \$v, I want to compute the intersection of the closed halfspaces given by the affine hull of each incident facet to \$v and such that those halfspaces do not contain the polytope.

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```> print \$c->FACETS->minor(\$c->FACETS_THRU_VERTICES->,All); 1 1 0 0 1 0 1 0 1 0 0 1 ```
minor(\$c->FACETS_THRU_VERTICES->,All) was actually the function I was looking for. However when I will compute \$C, I think I should consider the opposite normal vector of each the facet, since I don't want the polytope \$c to be contained in the intersection. Or maybe I do not get the real difference between FACETS and INEQUALITIES?