smallestFace -- determines the smallest face of the Cone/Polyhedron containing a point

Synopsis

• Usage:

  C1 = smallestFace(p,C)

  P1 = smallestFace(p,P)

• Inputs:
  • p, a matrix, over ZZ or QQ with only one column representing a point
  • C, a convex rational cone
  • P, a convex polyhedron
• Outputs:
  • C1, a convex rational cone, or
  • P1, a convex polyhedron

Description

p is considered to be a point in the ambient space of the second argument, so the number of rows of p must equal the dimension of the ambient space of the second argument. The function computes the smallest face of the second argument that contains p. If the second argument is a Polyhedron the output is a Polyhedron and if it is a Cone the output is a Cone. In both cases, if the point is not contained in the second argument then the output is the empty polyhedron.

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 : p = matrix {{1},{0},{0}}

o2 = | 1 |
     | 0 |
     | 0 |

              3        1
o2 : Matrix ZZ  <--- ZZ

i3 : smallestFace(p,P)

o3 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 2
      number of facets => 4
      number of rays => 0
      number of vertices => 4

o3 : Polyhedron