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Why is this polar polytope incorrect?

asked 2016-09-18 18:23:35 -0500

done_with_fish gravatar image

updated 2016-09-18 18:26:59 -0500

I have a polytope

sage: P = Polyhedron([
....: [-1, -1, -1, -1, -1], 
....: [-1, -1, -1, -1, 6], 
....: [-1, -1, -1, 6, -1], 
....: [-1, -1, 6, -1, -1], 
....: [-1, 6, -1, -1, -1], 
....: [5/2, -1, -1, -1, -1]
....: ])

The polar can be constructed manually with

sage: Pd = Polyhedron(ieqs=[vector([1]+v)for v in P.vertices_list()])
sage: Pd.vertices()
(A vertex at (1, 0, 0, 0, 0), 
 A vertex at (0, 1, 0, 0, 0),
 A vertex at (-2, -1, -1, -1, -1), 
 A vertex at (0, 0, 0, 0, 1), 
 A vertex at (0, 0, 0, 1, 0), 
 A vertex at (0, 0, 1, 0, 0))

This yields a result different from P.polar()

sage: P.polar().vertices()
(A vertex at (12/7, 6/7, 6/7, 6/7, 6/7), 
 A vertex at (0, 0, 0, 0, -6/7), 
 A vertex at (-12/7, 0, 0, 0, 0), 
 A vertex at (0, 0, 0, -6/7, 0), 
 A vertex at (0, -6/7, 0, 0, 0), 
 A vertex at (0, 0, -6/7, 0, 0))

Why am I getting different results? Is this a bug?

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answered 2016-09-20 03:13:47 -0500

Sébastien gravatar image

I am not an expert of polytopes, but I see that the documentation of P.polar() says that the "original vertices are translated so that their barycenter is at the origin".

I can confirm this by doing:

sage: Pcentred = P - P.center()
sage: Pcentred = - Pcentred           # I don't know why, but Sage .polar() method does this
sage: Pd = Polyhedron(ieqs=[vector([1]+v) for v in Pcentred.vertices_list()])
sage: Pd.vertices()
(A vertex at (12/7, 6/7, 6/7, 6/7, 6/7),
 A vertex at (0, 0, 0, 0, -6/7),
 A vertex at (-12/7, 0, 0, 0, 0),
 A vertex at (0, 0, 0, -6/7, 0),
 A vertex at (0, -6/7, 0, 0, 0),
 A vertex at (0, 0, -6/7, 0, 0))
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Comments

I guess this has the advantage of working for any polytope (the polar is only defined for polytopes containing the origin, I think).

done_with_fish gravatar imagedone_with_fish ( 2016-09-20 09:47:46 -0500 )edit

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Asked: 2016-09-18 18:23:35 -0500

Seen: 42 times

Last updated: Sep 20 '16