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isPure -- checks if a Fan is of pure dimension

Synopsis

Description

isPure tests if the Fan is pure by checking if the first and the last entry in the list of generating Cones are of the same dimension.

Let us construct a fan consisting of the positive orthant and the ray v that is the negative sum of the canonical basis, which is obviously not pure:

i1 : C = posHull matrix {{1,0,0},{0,1,0},{0,0,1}}

o1 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of the cone => 3
      number of facets => 3
      number of rays => 3

o1 : Cone
i2 : v = posHull matrix {{-1},{-1},{-1}}

o2 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of the cone => 1
      number of facets => 1
      number of rays => 1

o2 : Cone
i3 : F = fan {C,v}

o3 = {ambient dimension => 3         }
      number of generating cones => 2
      number of rays => 4
      top dimension of the cones => 3

o3 : Fan
i4 : isPure F

o4 = true

But we can make a pure fan if we choose any two dimensional face of the positive orthant and take the cone generated by this face and v and add it to the cone:

i5 : C1 = posHull{(faces(1,C))#0,v}

o5 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of the cone => 3
      number of facets => 3
      number of rays => 3

o5 : Cone
i6 : F = addCone(C1,F)

o6 = {ambient dimension => 3         }
      number of generating cones => 2
      number of rays => 4
      top dimension of the cones => 3

o6 : Fan
i7 : isPure F

o7 = true

Ways to use isPure :