Hi,

I am trying to use the "Polyhedra" package for computing with polyhedra. It has a function "polarFace", which returns the face F* of P* (the polar polytope of P) dual to F of P. However, in all dimensions, the returned polytope is just a point. Here is an example where this fails:

V = projectiveSpace(4)
KX= fold(plus,toList apply(0..4, i-> V_i))
P =  polytope KX -- this is just the polytope corresponding whose normal fan is the fan of P^4
F = first faces(2,P)
i139 : polarFace F

o139 = {ambient dimension => 4           }
        dimension of lineality space => 0
        dimension of polyhedron => -1
        number of facets => 0
        number of rays => 0
        number of vertices => 0



Clearly, F should have dimension 2 and not be the empty polyhedron. Some investigation in the source code reveals that the function "faces" fails to update F.cache to include the information that it is a face of P, and hence the empty polyhedron is returned.

Looking at the documentation, one sees that the bug is even present in the example run (http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9/share/doc/Macaulay2/Polyhedra/html/_polar__Face.html).

Am I doing something wrong, or is this really a bug?

Best,
Fredrik Meyer

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Seth Sullivant 	
Jun 28 (21 hours ago)
I was trying to use the polyhedra package to compute mixed volumes with the function "mixedVolume"

loadPackage "Polyhedra"
M1 = matrix{{0,0,0,0,0},{1,1,0,0,0},{1,2,0,0,0}}
P1 = convexHull transpose M1
M2 = matrix{{0,0,0,0,0},{1,0,1,0,0},{1,0,2,0,0}}
P2 = convexHull transpose M2
M3 = matrix{{0,0,0,0,0},{1,0,0,1,0},{1,0,0,2,0}}
P3 = convexHull transpose M3
M4 = matrix{{0,0,0,0,0},{1,0,0,0,1},{1,0,0,0,2}}
P4 = convexHull transpose M4
M5 = matrix{{0,1,1,0,0},{0,0,0,1,1}}
P5 = convexHull transpose M5
mixedVolume({P1,P2, P3,P4, P5})

I computed this 4 times and got four different answers: 0, 1, 4, and 5.  The correct value of the mixed volume in this case is 6 (computed with  PHCPack).  Josephine Yu told me that the problem is probably with the randomization that goes in to a certain lifting for the polyhedral computation (only uses random integers between 0 and 24). 

I'm not sure if there is anything to be done at this point, but I wanted to make people that might look aware of this problem.