chap6.txt

  
  6 Polytope
  
  
  6.1 Polytope: Category and Representations
  
  6.1-1 IsPolytope
  
  IsPolytope( M )  Category
  Returns:  true or false
  
  The GAP category of a polytope. Every polytope is a convex object.
  
  Remember: Every cone is a convex object.
  
  
  6.2 Polytope: Properties
  
  6.2-1 IsNotEmpty
  
  IsNotEmpty( poly )  property
  Returns:  true or false
  
  Checks if the polytope poly is not empty.
  
  6.2-2 IsLatticePolytope
  
  IsLatticePolytope( poly )  property
  Returns:  true or false
  
  Checks if the polytope poly is a lattice polytope, i.e. all its vertices are
  lattice points.
  
  6.2-3 IsVeryAmple
  
  IsVeryAmple( poly )  property
  Returns:  true or false
  
  Checks if the polytope poly is very ample.
  
  6.2-4 IsNormalPolytope
  
  IsNormalPolytope( poly )  property
  Returns:  true or false
  
  Checks if the polytope poly is normal.
  
  6.2-5 IsSimplicial
  
  IsSimplicial( poly )  property
  Returns:  true or false
  
  Checks if the polytope poly is simplicial.
  
  6.2-6 IsSimplePolytope
  
  IsSimplePolytope( poly )  property
  Returns:  true or false
  
  Checks if the polytope poly is simple.
  
  
  6.3 Polytope: Attributes
  
  6.3-1 Vertices
  
  Vertices( poly )  attribute
  Returns:  a list
  
  Returns  the  vertices  of the polytope poly. For reasons, the corresponding
  tester is HasVerticesOfPolytopes
  
  6.3-2 LatticePoints
  
  LatticePoints( poly )  attribute
  Returns:  a list
  
  Returns the lattice points of the polytope poly.
  
  6.3-3 FacetInequalities
  
  FacetInequalities( poly )  attribute
  Returns:  a list
  
  Returns the facet inequalities for the polytope poly.
  
  6.3-4 VerticesInFacets
  
  VerticesInFacets( poly )  attribute
  Returns:  a list
  
  Returns the incidence matrix of vertices and facets of the polytope poly.
  
  6.3-5 AffineCone
  
  AffineCone( poly )  attribute
  Returns:  a cone
  
  Returns the affine cone of the polytope poly.
  
  6.3-6 NormalFan
  
  NormalFan( poly )  attribute
  Returns:  a fan
  
  Returns the normal fan of the polytope poly.
  
  6.3-7 RelativeInteriorLatticePoints
  
  RelativeInteriorLatticePoints( poly )  attribute
  Returns:  a list
  
  Returns the lattice points in the relative interior of the polytope poly.
  
  
  6.4 Polytope: Methods
  
  6.4-1 *
  
  *( polytope1, polytope2 )  operation
  Returns:  a polytope
  
  Returns the Cartesian product of the polytopes polytope1 and polytope2.
  
  6.4-2 #
  
  #( polytope1, polytope2 )  operation
  Returns:  a polytope
  
  Returns the Minkowski sum of the polytopes polytope1 and polytope2.
  
  
  6.5 Polytope: Constructors
  
  6.5-1 Polytope
  
  Polytope( points )  operation
  Returns:  a polytope
  
  Returns a polytope that is the convex hull of the points points.
  
  6.5-2 PolytopeByInequalities
  
  PolytopeByInequalities( ineqs )  operation
  Returns:  a polytope
  
  Returns a polytope defined by the inequalities ineqs.
  
  
  6.6 Polytope: Examples
  
  
  6.6-1 Polytope example
  
    Example  
    gap> P := Polytope( [ [ 2, 0 ], [ 0, 2 ], [ -1, -1 ] ] );
    <A polytope in |R^2>
    gap> IsVeryAmple( P );
    true
    gap> LatticePoints( P );
    [ [ -1, -1 ], [ 0, 0 ], [ 0, 1 ], 
    [ 0, 2 ], [ 1, 0 ], [ 1, 1 ], [ 2, 0 ] ]
    gap> NFP := NormalFan( P );
    <A complete fan in |R^2>
    gap> C1 := MaximalCones( NFP )[ 1 ];
    <A cone in |R^2>
    gap> RayGenerators( C1 );
    [ [ -1, -1 ], [ -1, 3 ] ]
    gap> IsRegularFan( NFP );
    true