  
  [1X6 [33X[0;0YPolytope[133X[101X
  
  
  [1X6.1 [33X[0;0YPolytope: Category and Representations[133X[101X
  
  [1X6.1-1 IsPolytope[101X
  
  [33X[1;0Y[29X[2XIsPolytope[102X( [3XM[103X ) [32X Category[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YThe [5XGAP[105X category of a polytope. Every polytope is a convex object.[133X
  
  [33X[0;0YRemember: Every cone is a convex object.[133X
  
  
  [1X6.2 [33X[0;0YPolytope: Properties[133X[101X
  
  [1X6.2-1 IsNotEmpty[101X
  
  [33X[1;0Y[29X[2XIsNotEmpty[102X( [3Xpoly[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the polytope [3Xpoly[103X is not empty.[133X
  
  [1X6.2-2 IsLatticePolytope[101X
  
  [33X[1;0Y[29X[2XIsLatticePolytope[102X( [3Xpoly[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the polytope [3Xpoly[103X is a lattice polytope, i.e. all its vertices are
  lattice points.[133X
  
  [1X6.2-3 IsVeryAmple[101X
  
  [33X[1;0Y[29X[2XIsVeryAmple[102X( [3Xpoly[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the polytope [3Xpoly[103X is very ample.[133X
  
  [1X6.2-4 IsNormalPolytope[101X
  
  [33X[1;0Y[29X[2XIsNormalPolytope[102X( [3Xpoly[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the polytope [3Xpoly[103X is normal.[133X
  
  [1X6.2-5 IsSimplicial[101X
  
  [33X[1;0Y[29X[2XIsSimplicial[102X( [3Xpoly[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the polytope [3Xpoly[103X is simplicial.[133X
  
  [1X6.2-6 IsSimplePolytope[101X
  
  [33X[1;0Y[29X[2XIsSimplePolytope[102X( [3Xpoly[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X or [10Xfalse[110X[133X
  
  [33X[0;0YChecks if the polytope [3Xpoly[103X is simple.[133X
  
  
  [1X6.3 [33X[0;0YPolytope: Attributes[133X[101X
  
  [1X6.3-1 Vertices[101X
  
  [33X[1;0Y[29X[2XVertices[102X( [3Xpoly[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns  the  vertices  of the polytope [3Xpoly[103X. For reasons, the corresponding
  tester is HasVerticesOfPolytopes[133X
  
  [1X6.3-2 LatticePoints[101X
  
  [33X[1;0Y[29X[2XLatticePoints[102X( [3Xpoly[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns the lattice points of the polytope [3Xpoly[103X.[133X
  
  [1X6.3-3 FacetInequalities[101X
  
  [33X[1;0Y[29X[2XFacetInequalities[102X( [3Xpoly[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns the facet inequalities for the polytope [3Xpoly[103X.[133X
  
  [1X6.3-4 VerticesInFacets[101X
  
  [33X[1;0Y[29X[2XVerticesInFacets[102X( [3Xpoly[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns the incidence matrix of vertices and facets of the polytope [3Xpoly[103X.[133X
  
  [1X6.3-5 AffineCone[101X
  
  [33X[1;0Y[29X[2XAffineCone[102X( [3Xpoly[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya cone[133X
  
  [33X[0;0YReturns the affine cone of the polytope [3Xpoly[103X.[133X
  
  [1X6.3-6 NormalFan[101X
  
  [33X[1;0Y[29X[2XNormalFan[102X( [3Xpoly[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya fan[133X
  
  [33X[0;0YReturns the normal fan of the polytope [3Xpoly[103X.[133X
  
  [1X6.3-7 RelativeInteriorLatticePoints[101X
  
  [33X[1;0Y[29X[2XRelativeInteriorLatticePoints[102X( [3Xpoly[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns the lattice points in the relative interior of the polytope [3Xpoly[103X.[133X
  
  
  [1X6.4 [33X[0;0YPolytope: Methods[133X[101X
  
  [1X6.4-1 *[101X
  
  [33X[1;0Y[29X[2X*[102X( [3Xpolytope1[103X, [3Xpolytope2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya polytope[133X
  
  [33X[0;0YReturns the Cartesian product of the polytopes [3Xpolytope1[103X and [3Xpolytope2[103X.[133X
  
  [1X6.4-2 #[101X
  
  [33X[1;0Y[29X[2X#[102X( [3Xpolytope1[103X, [3Xpolytope2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya polytope[133X
  
  [33X[0;0YReturns the Minkowski sum of the polytopes [3Xpolytope1[103X and [3Xpolytope2[103X.[133X
  
  
  [1X6.5 [33X[0;0YPolytope: Constructors[133X[101X
  
  [1X6.5-1 Polytope[101X
  
  [33X[1;0Y[29X[2XPolytope[102X( [3Xpoints[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya polytope[133X
  
  [33X[0;0YReturns a polytope that is the convex hull of the points [3Xpoints[103X.[133X
  
  [1X6.5-2 PolytopeByInequalities[101X
  
  [33X[1;0Y[29X[2XPolytopeByInequalities[102X( [3Xineqs[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya polytope[133X
  
  [33X[0;0YReturns a polytope defined by the inequalities [3Xineqs[103X.[133X
  
  
  [1X6.6 [33X[0;0YPolytope: Examples[133X[101X
  
  
  [1X6.6-1 [33X[0;0YPolytope example[133X[101X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XP := Polytope( [ [ 2, 0 ], [ 0, 2 ], [ -1, -1 ] ] );[127X[104X
    [4X[28X<A polytope in |R^2>[128X[104X
    [4X[25Xgap>[125X [27XIsVeryAmple( P );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLatticePoints( P );[127X[104X
    [4X[28X[ [ -1, -1 ], [ 0, 0 ], [ 0, 1 ], [128X[104X
    [4X[28X[ 0, 2 ], [ 1, 0 ], [ 1, 1 ], [ 2, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XNFP := NormalFan( P );[127X[104X
    [4X[28X<A complete fan in |R^2>[128X[104X
    [4X[25Xgap>[125X [27XC1 := MaximalCones( NFP )[ 1 ];[127X[104X
    [4X[28X<A cone in |R^2>[128X[104X
    [4X[25Xgap>[125X [27XRayGenerators( C1 );[127X[104X
    [4X[28X[ [ -1, -1 ], [ -1, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsRegularFan( NFP );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
