normalCone(Polyhedron,Polyhedron) -- computes the normal cone of a face of a polyhedron

Synopsis

• Function: normalCone

• Usage:

  C = normalCone(P,Q)

• Inputs:
  • P, a convex polyhedron
  • Q, a convex polyhedron, which must be a face of P
• Optional inputs:
  • BasisElementLimit => ..., default value infinity,
  • DegreeLimit => ..., default value {},
  • MinimalGenerators => ..., default value true,
  • PairLimit => ..., default value infinity,
  • Strategy => ..., default value null,
  • Variable => ..., default value w,
• Outputs:
  • C, a convex rational cone

Description

The normal cone of a face Q of a polyhedron P is the cone in the normal fan (see normalFan) that corresponds to this face. This is the cone of all vectors attaining their maximum on this face.

i1 : P = hypercube 3

o1 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 3
      number of facets => 6
      number of rays => 0
      number of vertices => 8

o1 : Polyhedron

i2 : Q = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1},{1,1,1,1}}

o2 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 2
      number of facets => 4
      number of rays => 0
      number of vertices => 4

o2 : Polyhedron

i3 : C = normalCone(P,Q)

o3 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of the cone => 1
      number of facets => 1
      number of rays => 1

o3 : Cone

i4 : rays C

o4 = | 0  |
     | 0  |
     | -1 |

              3        1
o4 : Matrix ZZ  <--- ZZ