Strugles with VERTICES_IN_FACETS

[rado.rakotonarivo]

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]

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->[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).

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> print $c->FACETS_THRU_VERTICES->[0]; {0 2 4}

If you want to get the inequalities:

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> 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:

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$C=new Polytope(FACETS=>$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]

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?