Polyhedra : Table of Contents

• Polyhedra -- for computations with convex polyhedra, cones, and fans
• addCone -- adds cones to a Fan
• addPolyhedron -- adds Polyhedra to a PolyhedralComplex
• affineHull -- computes the affine hull of a polyhedron
• affineImage -- computes the affine image of a cone or polyhedron
• affineImage(Matrix,Cone,Matrix) -- computes the affine image of a cone
• affineImage(Matrix,Polyhedron,Matrix) -- computes the affine image of a polyhedron
• affinePreimage -- computes the affine preimage of a cone or polyhedron
• affinePreimage(Matrix,Cone,Matrix) -- computes the affine preimage of a cone
• affinePreimage(Matrix,Polyhedron,Matrix) -- computes the affine preimage of a polyhedron
• ambDim -- ambient dimension of a Polyhedron, Cone or Fan
• areCompatible -- checks if the intersection of two cones/polyhedra is a face of each
• barycentricTriangulation -- computes a triangulation of a polytope
• bipyramid -- computes the bipyramid over a polyhedron
• ccRefinement -- computes the coarsest common refinement of a set of rays
• cellDecompose -- Deprecated variant of {\tt regularSubdivision}
• commonFace -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans
• Cone -- the class of all rational convex polyhedral cones
• Cone * Cone -- computes the direct product of two cones
• Cone * Polyhedron -- computes the direct product of a cone and a polyhedron
• Cone + Cone -- computes the Minkowski sum of two cones
• Cone + Polyhedron -- computes the Minkowski sum of a cone and a polyhedron
• Cone == Cone -- equality
• Cone ? Cone -- compares the Cones
• cone(Polyhedron) -- Take the cone over a polyhedron
• coneFromHData -- Constructing a polyhedral cone as intersection of halfspaces.
• coneFromVData -- computes the positive hull of rays, cones, and the cone over a polyhedron
• cones -- computes all cones of a fan of a certain dimension
• contains -- checks if the first argument contains the second argument
• convexHull -- computing the convex hull of points, rays and polyhedra
• crossPolytope -- computes the d-dimensional crosspolytope with diameter 2s
• cyclicPolytope -- computes the d dimensional cyclic polytope with n vertices
• dim(PolyhedralObject) -- computes the dimension of a cone, polyhedron, fan or polyhedral complex
• directProduct -- computes the direct product of two convex objects
• directProduct(Cone,Cone) -- computes the direct product of polyhedra and cones
• directProduct(Fan,Fan) -- computes the direct product of two fans
• dualCone -- computes the dual Cone
• dualFaceRepresentationMap (missing documentation)
• ehrhart -- calculates the Ehrhart polynomial of a polytope
• emptyPolyhedron -- generates the empty polyhedron in n-space
• faceFan -- computes the fan generated by the cones over the faces
• faces -- computes all faces of a certain codimension of a Cone or Polyhedron
• faces(PolyhedralObject) -- Giving the faces of a polyhedral object.
• facesAsCones -- Returns the faces of a cone as actual cones.
• facesAsPolyhedra -- Returns the faces of a polyhedron as actual polyhedra.
• facets -- Giving the facet inequalities of a cone or polyhedron.
• Fan -- the class of all fans
• fan -- generates a Fan
• Fan * Fan -- computes the direct product
• Fan == Fan -- equality
• fan(Matrix,Matrix,List) -- Constructing a fan.
• fan(PolyhedralComplex) -- Take the fan over a polyhedral complex
• fanFromGfan -- Construct a fan from output data of {\tt Gfan}
• fVector -- computes the f-vector of a Cone, Polyhedron, Fan or PolyhedralComplex
• halfspaces -- computes the defining half-spaces of a Cone or a Polyhedron
• hilbertBasis -- computes the Hilbert basis of a Cone
• hirzebruch -- computes the fan of the r-th Hirzebruch surface
• hypercube -- Returns the d-dimensional hypercube
• hyperplanes -- computes the defining hyperplanes of a Cone or a Polyhedron
• imageFan -- computes the fan of the image
• incompCones -- returns the pairs of incompatible cones
• incompPolyhedra -- returns the pairs of incompatible polyhedra
• inInterior -- checks if a point lies in the relative interior of a Cone/Polyhedron
• interiorLatticePoints -- computes the lattice points in the relative interior of a polytope
• interiorPoint -- computes a point in the relative interior of the Polyhedron
• interiorVector -- computes a vector in the relative interior of a Cone
• intersection -- computes the intersection of cones, and polyhedra
• intersection(Matrix) -- Deprecated variant of {\tt coneFromHData}
• intersection(Matrix,Matrix) -- Deprecated variants of {\tt polyhedronFromHData} and {\tt coneFromHData}
• isCompact -- checks compactness of a Polyhedron
• isComplete -- checks completeness of a Fan or PolyhedralComplex
• isEmpty -- checks if a Polyhedron is empty
• isFace -- tests if the first argument is a face of the second
• isFullDimensional -- Determine whether a polyhedral object is full-dimensional
• isLatticePolytope -- checks if a polyhedron is a lattice polytope
• isNormal(Polyhedron) -- checks if a polytope is normal in the ambient lattice
• isPointed -- checks if a Cone or Fan is pointed
• isPolytopal -- checks if a Fan is polytopal
• isPure -- checks if a Fan or PolyhedralComplex is of pure dimension
• isReflexive -- checks if a Polytope is reflexive
• isSimplicial -- checks if a polyhedral object is simplicial
• isSmooth(Cone) -- checks if a Cone or Fan is smooth
• isVeryAmple(Polyhedron) -- checks if the Polyhedron is very ample
• isWellDefined(Cone) -- Checks whether a polyhedral object is well-defined.
• latticePoints -- computes the lattice points of a polytope
• latticeVolume -- Returning the lattice volume of a polyhedron.
• linealitySpace -- computes a basis of the lineality space
• linearTransform (missing documentation)
• linSpace -- Deprecated version of @TO "linealitySpace"@
• maxCones -- displays the generating Cones of a Fan
• maxFace -- computes the face of a Polyhedron or Cone where a weight attains its maximum
• maxPolyhedra -- displays the generating Polyhedra of a PolyhedralComplex
• minFace -- computes the face of a Polyhedron or Cone where a weight attains its minimum
• minimalNonFaces (missing documentation)
• minimalNonFaces(Fan) -- Giving the minimal non-faces of a fan..
• minkowskiSum -- computes the Minkowski sum of two convex objects
• minkSummandCone -- computes the Cone of all Minkowski summands and the minimal decompositions
• mixedVolume -- computes the mixed volume of a list of polytope
• newtonPolytope -- computes the Newton polytope of a polynomial
• normalCone(Polyhedron,Polyhedron) -- computes the normal cone of a face of a polyhedron
• normalFan -- computes the normalFan of a polyhedron
• nVertices -- Returns the number of vertices of a polyhedron
• objectiveVector -- computes an objective vector of a face of a polyhedron
• polar -- computes the polar of a polyhedron
• polarFace -- computes the dual face of the polar polyhedron
• polyhedra -- computes all polyhedra of a polyhedral complex of a certain dimension
• PolyhedralComplex -- the class of all polyhedral complexes
• polyhedralComplex -- generates a PolyhedralComplex
• polyhedralComplex(Fan) -- Turn a fan into a polyhedral complex
• polyhedralComplex(Matrix,Matrix,Matrix,List) -- Constructing a polyhedral complex.
• PolyhedralObject -- the class of all polyhedral objects in Polyhedra
• Polyhedron -- the class of all convex polyhedra
• polyhedron -- Turn a cone into a polyhedron
• Polyhedron * Cone -- computes the direct product of a polyhedron and a cone
• Polyhedron * Polyhedron -- computes the direct product of two polyhedra
• Polyhedron + Cone -- computes the Minkowski sum of a polyhedron and a cone
• Polyhedron + Polyhedron -- computes the Minkowski sum of two polyhedra
• Polyhedron == Polyhedron -- equality
• polyhedronFromHData -- Constructing a polyhedron from its H-representation, i.e. inequalities and equations
• polytope -- returns a polytope of which the fan is the normal fan if it is polytopal
• posOrthant -- generates the positive orthant in n-space
• proximum -- computes the proximum of the Polyhedron/Cone to a point in euclidean metric
• pyramid -- computes the pyramid over a polyhedron
• QQ * Polyhedron -- rescales a polyhedron by a given positive factor
• rays(PolyhedralObject) -- displays all rays of a Cone, a Fan, or a Polyhedron
• regularSubdivision -- Computes the regular cell decomposition
• regularTriangulation -- Computes a regular triangulation of a given polytope.
• saveSession -- save the actual Polyhedra session to a file
• secondaryPolytope -- computes the secondary polytope of a compact polyhedron
• simplex -- Produces a full-dimensional simplex
• skeleton -- computes the k-skeleton of a Fan or PolyhedralComplex
• smallestFace -- determines the smallest face of the Cone/Polyhedron containing a point
• smoothSubfan -- computes the subfan of all smooth cones
• stanleyReisnerRing (missing documentation)
• stanleyReisnerRing(Fan) -- Give the Stanley–Reisner ring of a fan.
• statePolytope -- computes the state polytope of a homogeneous ideal
• stdSimplex -- generates the d-dimensional standard simplex
• stellarSubdivision -- computes the stellar subdivision of the fan by a ray
• sublatticeBasis -- computes a basis for the sublattice generated by integral vectors or lattice points of a polytope
• tailCone -- computes the tail/recession cone of a polyhedron
• toSublattice -- calculates the preimage of a polytope in the sublattice generated by its lattice points
• triangulate -- Deprecated name for {\tt barycentricTriangulation}
• V- and H-representation
• vertexEdgeMatrix -- computes the vertex-edge-relations matrix
• vertexFacetMatrix -- computes the vertex-facet-relations matrix
• vertices -- displays the vertices of a Polyhedron or a PolyhedralComplex
• volume -- computes the volume of a polytope
• Working with cones
• Working with fans
• Working with fans - Part 2
• Working with polyhedra