# Polyhedra -- for computations with convex polyhedra, cones, and fans

## Description

A rational convex Polyhedron is the intersection of finitely many affine half-spaces over QQ or equivalently, the convex hull of a finite set of vertices and rays. A rational convex polyhedral Cone is the intersection of finitely many linear half-spaces over QQ or equivalently, the positive hull of a finite set of rays. A Fan is a finite collection of cones such that for each cone all its faces are in the fan and for two cones in the fan the intersection is a face of each.

Polyhedra uses the

Here are some examples illustrating the main uses of this package.

Please have a look at V- and H-representation on conventions used throughout this package before you start.

For an introduction to polyhedra and cones, we recommend Gunter M. Ziegler's Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995.

The authors would like to thank

for code contributions, bug reports, and helpful discussions concerning this package.

## Certification

Version 1.0.5 of this package was accepted for publication in volume 1 of The Journal of Software for Algebra and Geometry: Macaulay2 on 2009-09-07, in the article Polyhedra: a package for computations with convex polyhedral objects. That version can be obtained from the journal or from the Macaulay2 source code repository.

## Version

This documentation describes version 1.10 of Polyhedra.

## Source code

The source code from which this documentation is derived is in the file Polyhedra.m2. The auxiliary files accompanying it are in the directory Polyhedra/.

## Exports

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

## For the programmer

The object Polyhedra is .