inInterior -- checks if a point lies in the relative interior of a Cone/Polyhedron

Usage:

b = inInterior(p,C)

b = inInterior(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:
- b, a Boolean value, true if p lies in the relative interior of the Cone/Polyhedron, false otherwise

Description

inInterior checks if the smallest face of the Cone or the Polyhedron containing p is the Cone or the Polyhedron itself. For this the number of rows of p must equal the ambient dimension of the second argument.

i1 : P = cyclicPolytope(3,5)

o1 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 3
      number of facets => 6
      number of rays => 0
      number of vertices => 5

o1 : Polyhedron

i2 : p = matrix{{2},{4},{8}}

o2 = | 2 |
     | 4 |
     | 8 |

              3       1
o2 : Matrix ZZ  <-- ZZ

i3 : q = matrix{{2},{6},{20}}

o3 = | 2  |
     | 6  |
     | 20 |

              3       1
o3 : Matrix ZZ  <-- ZZ

i4 : inInterior(p,P)

o4 = false

i5 : inInterior(q,P)

o5 = true

Ways to use inInterior:

inInterior(Matrix,Cone)
inInterior(Matrix,Polyhedron)

For the programmer

The object inInterior is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldPolyhedra.m2:6249:0.