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

Ways to use this method: