## Strugles with VERTICES_IN_FACETS

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rado.rakotonarivo
Posts: 2
Joined: 06 Feb 2019, 16:59

### Strugles with VERTICES_IN_FACETS

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

joswig
Main Author
Posts: 216
Joined: 24 Dec 2010, 11:10

### Re: Strugles with VERTICES_IN_FACETS

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 $c. Sorry, I do not understand what you want to compute. Do you want to compute the intersection of the affine halfspaces given by the facets through a fixed vertex? This would be like the following. Code: Select all >$c=cube(3); > print $c->VERTICES->[0]; 1 -1 -1 -1  Just to check which vertex is indexed with 0. Here you see the indices of the facets containing that vertices (as row indices to the FACETS matrix). Code: Select all > print$c->FACETS_THRU_VERTICES->[0]; {0 2 4} 
If you want to get the inequalities:

Code: Select all

> print $c->FACETS->minor($c->FACETS_THRU_VERTICES->[0],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:

Code: Select all

$C=new Polytope(FACETS=>$c->FACETS->minor($c->FACETS_THRU_VERTICES->[0],All));  Here using FACETS (instead of the standard INEQUALITIES) as an input property is OK, because you know that there are no hidden redundancies. rado.rakotonarivo Posts: 2 Joined: 06 Feb 2019, 16:59 ### Re: Strugles with VERTICES_IN_FACETS Thank you for your fast reply, Sorry, I do not understand what you want to compute. Do you want to compute the intersection of the affine halfspaces given by the facets through a fixed vertex? This is indeed what I aim for but I want to consider the closed halfspaces that does not contain the polytope. That is, for a fixed vertex$v, I want to compute the intersection of the closed halfspaces given by the affine hull of each incident facet to $v and such that those halfspaces do not contain the polytope. Code: Select all > print$c->FACETS->minor($c->FACETS_THRU_VERTICES->[0],All); 1 1 0 0 1 0 1 0 1 0 0 1  minor($c->FACETS_THRU_VERTICES->[0],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?

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