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<head><title>Polyhedra -- for computations with convex polyhedra, cones, and fans</title>
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<div><h1>Polyhedra -- for computations with convex polyhedra, cones, and fans</h1>
<div class="single"><h2>Description</h2>
<div>A rational convex <a href="___Polyhedron.html" title="the class of all convex polyhedra">Polyhedron</a> is the intersection of finitely many affine half-spaces
over <a href="../../Macaulay2Doc/html/___Q__Q.html" title="the class of all rational numbers">QQ</a> or equivalently, the convex hull of a finite set of vertices and rays.
A rational convex polyhedral <a href="___Cone.html" title="the class of all rational convex polyhedral cones">Cone</a> is the intersection of finitely many linear half-spaces
over <a href="../../Macaulay2Doc/html/___Q__Q.html" title="the class of all rational numbers">QQ</a> or equivalently, the positive hull of a finite set of rays. A <a href="___Fan.html" title="the class of all fans">Fan</a> 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.<p/>
<tt>Polyhedra</tt> uses the <a href="../../FourierMotzkin/html/index.html" title="for convex hull and vertex enumeration">FourierMotzkin</a> package by <a href="http://www.mast.queensu.ca/~ggsmith">Gregory G. Smith</a>. Each polyhedron or cone is
saved in both descriptions and a fan is saved as the list of its generating cones.<p/>
Here are some examples illustrating the main uses of this package.<ul><li><a href="___Working_spwith_sppolyhedra.html" title="">Working with polyhedra</a></li>
<li><a href="___Working_spwith_spcones.html" title="">Working with cones</a></li>
<li><a href="___Working_spwith_spfans.html" title="">Working with fans</a></li>
</ul>
<p/>
For an introduction to polyhedra and cones, we recommend <a href="http://www.math.tu-berlin.de/~ziegler/">Gunter
M. Ziegler's</a> <em>Lectures on Polytopes</em>, Graduate
Texts in Mathematics 152, Springer-Verlag, New York, 1995.<p/>
The author would like to thank <a href="http://people.cs.uchicago.edu/~nilten/">Nathan Ilten</a>
for contributing several functions to the package.</div>
</div>
<div class="single"><h2>Author</h2>
<ul><li><div class="single"><a href="http://page.mi.fu-berlin.de/rbirkner/index.htm">René Birkner</a><span> <<a href="mailto:rbirkner@mi.fu-berlin.de">rbirkner@mi.fu-berlin.de</a>></span></div>
</li>
</ul>
</div>
<div class="single"><h2>Certification</h2>
<p>Version <b>1.0.5</b> of this package was accepted for publication in <a href="http://j-sag.org/Volume1/">volume 1</a> of the journal <a href="http://j-sag.org/">The Journal of Software for Algebra and Geometry: Macaulay2</a> on 2009-09-07, in the article <a href="http://j-sag.org/Volume1/jsag-3-2009.pdf">Polyhedra: a package for computations with convex polyhedral objects</a>. That version can be obtained <a href="http://j-sag.org/Volume1/Polyhedra.m2">from the journal</a> or from the <em>Macaulay2</em> source code repository, after installing <a href="http://subversion.tigris.org/">subversion</a>, with the following shell command:</p>
<pre> svn export -r 9344 svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/Polyhedra.m2</pre>
<p>The following command will display the log messages accompanying any changes to the file in the repository since publication.</p>
<pre> svn log -r 9345:HEAD svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/Polyhedra.m2</pre>
<p>The following command will summarize the changes to the file in the repository since publication, in the format the program <tt>diff</tt> uses: lines starting with <tt>+</tt> have been added, and lines starting with <tt>-</tt> have been removed. (Changes to white space or end of line style will not be reported.)</p>
<pre> svn diff -x "-b --ignore-eol-style" -r 9344:HEAD svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/Polyhedra.m2</pre>
<p>The differences between two releases in the repository mentioned in the log can be displayed by replacing <tt>9344:HEAD</tt> by the pair of release numbers separated by a colon.</p>
</div>
<div class="single"><h2>Version</h2>
This documentation describes version <b>1.0.8</b> of Polyhedra.</div>
<div class="single"><h2>Source code</h2>
The source code from which this documentation is derived is in the file <a href="../../../../Macaulay2/Polyhedra.m2">Polyhedra.m2</a>.</div>
<div class="single"><h2>Exports</h2>
<ul><li><div class="single">Types<ul><li><span><a href="___Cone.html" title="the class of all rational convex polyhedral cones">Cone</a> -- the class of all rational convex polyhedral cones</span></li>
<li><span><a href="___Fan.html" title="the class of all fans">Fan</a> -- the class of all fans</span></li>
<li><span><a href="___Polyhedral__Object.html" title="the class of all polyhedral objects in Polyhedra">PolyhedralObject</a> -- the class of all polyhedral objects in Polyhedra</span></li>
<li><span><a href="___Polyhedron.html" title="the class of all convex polyhedra">Polyhedron</a> -- the class of all convex polyhedra</span></li>
</ul>
</div>
</li>
<li><div class="single">Functions<ul><li><span><a href="_add__Cone.html" title="adds cones to a Fan">addCone</a> -- adds cones to a Fan</span></li>
<li><span><a href="_affine__Hull.html" title="computes the affine hull of a polyhedron">affineHull</a> -- computes the affine hull of a polyhedron</span></li>
<li><span><a href="_affine__Image.html" title="computes the affine image of a cone or polyhedron">affineImage</a> -- computes the affine image of a cone or polyhedron</span></li>
<li><span><a href="_affine__Preimage.html" title="computes the affine preimage of a cone or polyhedron">affinePreimage</a> -- computes the affine preimage of a cone or polyhedron</span></li>
<li><span><a href="_amb__Dim.html" title="ambient dimension of a Polyhedron, Cone or Fan">ambDim</a> -- ambient dimension of a Polyhedron, Cone or Fan</span></li>
<li><span><a href="_are__Compatible.html" title="checks if the intersection of two cones is a face of each">areCompatible</a> -- checks if the intersection of two cones is a face of each</span></li>
<li><span><a href="_bipyramid.html" title="computes the bipyramid over a polyhedron">bipyramid</a> -- computes the bipyramid over a polyhedron</span></li>
<li><span><a href="_cc__Refinement.html" title="computes the coarsest common refinement of a set of rays">ccRefinement</a> -- computes the coarsest common refinement of a set of rays</span></li>
<li><span><a href="_cell__Decompose.html" title="computes the regular cell decomposition">cellDecompose</a> -- computes the regular cell decomposition</span></li>
<li><span><a href="_common__Face.html" title="checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans">commonFace</a> -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans</span></li>
<li><span><a href="_cones.html" title="computes all cones of a fan of a certain dimension">cones</a> -- computes all cones of a fan of a certain dimension</span></li>
<li><span><a href="_cone__To__Polyhedron.html" title="converts a cone to class Polyhedron">coneToPolyhedron</a> -- converts a cone to class Polyhedron</span></li>
<li><span><a href="_contains.html" title="checks if the first argument contains the second argument">contains</a> -- checks if the first argument contains the second argument</span></li>
<li><span><a href="_convex__Hull.html" title="computing the convex hull of points, rays and polyhedra">convexHull</a> -- computing the convex hull of points, rays and polyhedra</span></li>
<li><span><a href="_cross__Polytope.html" title="computes the d-dimensional crosspolytope with diameter 2s">crossPolytope</a> -- computes the d-dimensional crosspolytope with diameter 2s</span></li>
<li><span><a href="_cyclic__Polytope.html" title="computes the d dimensional cyclic polytope with n vertices">cyclicPolytope</a> -- computes the d dimensional cyclic polytope with n vertices</span></li>
<li><span><a href="_direct__Product.html" title="computes the direct product of two convex objects">directProduct</a> -- computes the direct product of two convex objects</span></li>
<li><span><a href="_dual__Cone.html" title=" computes the dual Cone">dualCone</a> -- computes the dual Cone</span></li>
<li><span><a href="_dual__Face__Lattice.html" title="computes the dual face lattice of a cone or polyhedron">dualFaceLattice</a> -- computes the dual face lattice of a cone or polyhedron</span></li>
<li><span><a href="_ehrhart.html" title="calculates the Ehrhart polynomial of a polytope">ehrhart</a> -- calculates the Ehrhart polynomial of a polytope</span></li>
<li><span><a href="_empty__Polyhedron.html" title="generates the empty polyhedron in n-space">emptyPolyhedron</a> -- generates the empty polyhedron in n-space</span></li>
<li><span><a href="_face__Fan.html" title=" computes the fan generated by the cones over the faces">faceFan</a> -- computes the fan generated by the cones over the faces</span></li>
<li><span><a href="_face__Lattice.html" title="computes the face lattice of a cone or polyhedron">faceLattice</a> -- computes the face lattice of a cone or polyhedron</span></li>
<li><span><a href="_faces.html" title="computes all faces of a certain codimension of a Cone or Polyhedron">faces</a> -- computes all faces of a certain codimension of a Cone or Polyhedron</span></li>
<li><span><a href="_fan.html" title="generates a Fan">fan</a> -- generates a Fan</span></li>
<li><span><a href="_f__Vector.html" title="computes the f-vector of a Cone or Polyhedron">fVector</a> -- computes the f-vector of a Cone or Polyhedron</span></li>
<li><span><a href="_gen__Cones.html" title="displays the generating Cones of a Fan">genCones</a> -- displays the generating Cones of a Fan</span></li>
<li><span><a href="_halfspaces.html" title="computes the defining half-spaces of a Cone or a Polyhedron">halfspaces</a> -- computes the defining half-spaces of a Cone or a Polyhedron</span></li>
<li><span><a href="_hilbert__Basis.html" title="computes the Hilbert basis of a Cone">hilbertBasis</a> -- computes the Hilbert basis of a Cone</span></li>
<li><span><a href="_hirzebruch.html" title="computes the fan of the r-th Hirzebruch surface">hirzebruch</a> -- computes the fan of the r-th Hirzebruch surface</span></li>
<li><span><a href="_hypercube.html" title="computes the d-dimensional hypercube with edge length 2*s">hypercube</a> -- computes the d-dimensional hypercube with edge length 2*s</span></li>
<li><span><a href="_hyperplanes.html" title="computes the defining hyperplanes of a Cone or a Polyhedron">hyperplanes</a> -- computes the defining hyperplanes of a Cone or a Polyhedron</span></li>
<li><span><a href="_image__Fan.html" title=" computes the fan of the image">imageFan</a> -- computes the fan of the image</span></li>
<li><span><a href="_incomp__Cones.html" title="returns the pairs of incompatible cones">incompCones</a> -- returns the pairs of incompatible cones</span></li>
<li><span><a href="_in__Interior.html" title="checks if a point lies in the relative interior of a Cone/Polyhedron">inInterior</a> -- checks if a point lies in the relative interior of a Cone/Polyhedron</span></li>
<li><span><a href="_interior__Point.html" title="computes a point in the relative interior of the Polyhedron">interiorPoint</a> -- computes a point in the relative interior of the Polyhedron</span></li>
<li><span><a href="_interior__Vector.html" title="computes a vector in the relative interior of a Cone">interiorVector</a> -- computes a vector in the relative interior of a Cone</span></li>
<li><span><a href="_intersection.html" title="computes the intersection of half-spaces, hyperplanes, cones, and polyhedra">intersection</a> -- computes the intersection of half-spaces, hyperplanes, cones, and polyhedra</span></li>
<li><span><a href="_is__Compact.html" title="checks compactness of a Polyhedron">isCompact</a> -- checks compactness of a Polyhedron</span></li>
<li><span><a href="_is__Complete.html" title="checks completeness of a Fan">isComplete</a> -- checks completeness of a Fan</span></li>
<li><span><a href="_is__Empty.html" title="checks if a Polyhedron is empty">isEmpty</a> -- checks if a Polyhedron is empty</span></li>
<li><span><a href="_is__Face.html" title="tests if the first argument is a face of the second">isFace</a> -- tests if the first argument is a face of the second</span></li>
<li><span><a href="_is__Pointed.html" title="checks if a Cone or Fan is pointed">isPointed</a> -- checks if a Cone or Fan is pointed</span></li>
<li><span><a href="_is__Polytopal.html" title="checks if a Fan is polytopal">isPolytopal</a> -- checks if a Fan is polytopal</span></li>
<li><span><a href="_is__Pure.html" title="checks if a Fan is of pure dimension">isPure</a> -- checks if a Fan is of pure dimension</span></li>
<li><span><a href="_is__Smooth.html" title="checks if a Cone or Fan is smooth">isSmooth</a> -- checks if a Cone or Fan is smooth</span></li>
<li><span><a href="_lattice__Points.html" title="computes the lattice points of a polytope">latticePoints</a> -- computes the lattice points of a polytope</span></li>
<li><span><a href="_lin__Space.html" title="computes a basis of the lineality space">linSpace</a> -- computes a basis of the lineality space</span></li>
<li><span><a href="_max__Face.html" title="computes the face of a Polyhedron or Cone where a weight attains its maximum">maxFace</a> -- computes the face of a Polyhedron or Cone where a weight attains its maximum</span></li>
<li><span><a href="_min__Face.html" title="computes the face of a Polyhedron or Cone where a weight attains its minimum">minFace</a> -- computes the face of a Polyhedron or Cone where a weight attains its minimum</span></li>
<li><span><a href="_minkowski__Sum.html" title=" computes the Minkowski sum of two convex objects">minkowskiSum</a> -- computes the Minkowski sum of two convex objects</span></li>
<li><span><a href="_mink__Summand__Cone.html" title="computes the Cone of all Minkowski summands and the minimal decompositions">minkSummandCone</a> -- computes the Cone of all Minkowski summands and the minimal decompositions</span></li>
<li><span><a href="_newton__Polytope.html" title="computes the Newton polytope of a polynomial">newtonPolytope</a> -- computes the Newton polytope of a polynomial</span></li>
<li><span><a href="_normal__Fan.html" title="computes the normalFan of a polyhedron">normalFan</a> -- computes the normalFan of a polyhedron</span></li>
<li><span><a href="_objective__Vector.html" title="computes an objective vector of a face of a polyhedron">objectiveVector</a> -- computes an objective vector of a face of a polyhedron</span></li>
<li><span><a href="_polar.html" title=" computes the polar of a polyhedron">polar</a> -- computes the polar of a polyhedron</span></li>
<li><span><a href="_polytope.html" title="returns a polytope of which the fan is the normal fan if it is polytopal">polytope</a> -- returns a polytope of which the fan is the normal fan if it is polytopal</span></li>
<li><span><a href="_pos__Hull.html" title="computes the positive hull of rays, cones, and the cone over a polyhedron">posHull</a> -- computes the positive hull of rays, cones, and the cone over a polyhedron</span></li>
<li><span><a href="_pos__Orthant.html" title="generates the positive orthant in n-space">posOrthant</a> -- generates the positive orthant in n-space</span></li>
<li><span><a href="_proximum.html" title="computes the proximum of the Polyhedron/Cone to a point in euclidian metric">proximum</a> -- computes the proximum of the Polyhedron/Cone to a point in euclidian metric</span></li>
<li><span><a href="_pyramid.html" title="computes the pyramid over a polyhedron">pyramid</a> -- computes the pyramid over a polyhedron</span></li>
<li><span><a href="_rays.html" title="displays all rays of a Cone, a Fan, or a Polyhedron">rays</a> -- displays all rays of a Cone, a Fan, or a Polyhedron</span></li>
<li><span><a href="_save__Session.html" title="save the actual Polyhedra session to a file">saveSession</a> -- save the actual Polyhedra session to a file</span></li>
<li><span><a href="_secondary__Polytope.html" title="computes the secondary polytope of a compact polyhedron">secondaryPolytope</a> -- computes the secondary polytope of a compact polyhedron</span></li>
<li><span><a href="_skeleton.html" title="computes the k-skeleton of a Fan">skeleton</a> -- computes the k-skeleton of a Fan</span></li>
<li><span><a href="_smallest__Face.html" title="determines the smallest face of the Cone/Polyhedron containing a point">smallestFace</a> -- determines the smallest face of the Cone/Polyhedron containing a point</span></li>
<li><span><a href="_smooth__Subfan.html" title="computes the subfan of all smooth cones">smoothSubfan</a> -- computes the subfan of all smooth cones</span></li>
<li><span><a href="_state__Polytope.html" title="computes the state polytope of a homogeneous ideal">statePolytope</a> -- computes the state polytope of a homogeneous ideal</span></li>
<li><span><a href="_std__Simplex.html" title="generates the d-dimensional standard simplex">stdSimplex</a> -- generates the d-dimensional standard simplex</span></li>
<li><span><a href="_stellar__Subdivision.html" title="computes the stellar subdivision of the fan by a ray">stellarSubdivision</a> -- computes the stellar subdivision of the fan by a ray</span></li>
<li><span><a href="_sublattice__Basis.html" title="computes a basis for the sublattice generated by integral vectors or the lattice points of a polytope">sublatticeBasis</a> -- computes a basis for the sublattice generated by integral vectors or the lattice points of a polytope</span></li>
<li><span><a href="_tail__Cone.html" title="computes the tail/recession cone of a polyhedron">tailCone</a> -- computes the tail/recession cone of a polyhedron</span></li>
<li><span><a href="_to__Sublattice.html" title="calculates the preimage of a polytope in the sublattice generated by its lattice points">toSublattice</a> -- calculates the preimage of a polytope in the sublattice generated by its lattice points</span></li>
<li><span><a href="_triangulate.html" title="computes a triangulation of a polytope">triangulate</a> -- computes a triangulation of a polytope</span></li>
<li><span><a href="_vertex__Edge__Matrix.html" title="computes the vertex-edge-relations matrix">vertexEdgeMatrix</a> -- computes the vertex-edge-relations matrix</span></li>
<li><span><a href="_vertex__Facet__Matrix.html" title="computes the vertex-facet-relations matrix">vertexFacetMatrix</a> -- computes the vertex-facet-relations matrix</span></li>
<li><span><a href="_vertices.html" title="displays the vertices of a Polyhedron">vertices</a> -- displays the vertices of a Polyhedron</span></li>
<li><span><a href="_volume.html" title="computes the volume of a polytope">volume</a> -- computes the volume of a polytope</span></li>
</ul>
</div>
</li>
<li><div class="single">Methods<ul><li><span>addCone(Cone,Fan), see <span><a href="_add__Cone.html" title="adds cones to a Fan">addCone</a> -- adds cones to a Fan</span></span></li>
<li><span>addCone(Fan,Fan), see <span><a href="_add__Cone.html" title="adds cones to a Fan">addCone</a> -- adds cones to a Fan</span></span></li>
<li><span>addCone(List,Fan), see <span><a href="_add__Cone.html" title="adds cones to a Fan">addCone</a> -- adds cones to a Fan</span></span></li>
<li><span>affineImage(Cone,Matrix), see <span><a href="_affine__Image_lp__Matrix_cm__Cone_cm__Matrix_rp.html" title="computes the affine image of a cone">affineImage(Matrix,Cone,Matrix)</a> -- computes the affine image of a cone</span></span></li>
<li><span>affineImage(Matrix,Cone), see <span><a href="_affine__Image_lp__Matrix_cm__Cone_cm__Matrix_rp.html" title="computes the affine image of a cone">affineImage(Matrix,Cone,Matrix)</a> -- computes the affine image of a cone</span></span></li>
<li><span><a href="_affine__Image_lp__Matrix_cm__Cone_cm__Matrix_rp.html" title="computes the affine image of a cone">affineImage(Matrix,Cone,Matrix)</a> -- computes the affine image of a cone</span></li>
<li><span>affineImage(Matrix,Polyhedron), see <span><a href="_affine__Image_lp__Matrix_cm__Polyhedron_cm__Matrix_rp.html" title="computes the affine image of a polyhedron">affineImage(Matrix,Polyhedron,Matrix)</a> -- computes the affine image of a polyhedron</span></span></li>
<li><span><a href="_affine__Image_lp__Matrix_cm__Polyhedron_cm__Matrix_rp.html" title="computes the affine image of a polyhedron">affineImage(Matrix,Polyhedron,Matrix)</a> -- computes the affine image of a polyhedron</span></li>
<li><span>affineImage(Polyhedron,Matrix), see <span><a href="_affine__Image_lp__Matrix_cm__Polyhedron_cm__Matrix_rp.html" title="computes the affine image of a polyhedron">affineImage(Matrix,Polyhedron,Matrix)</a> -- computes the affine image of a polyhedron</span></span></li>
<li><span>affinePreimage(Cone,Matrix), see <span><a href="_affine__Preimage_lp__Matrix_cm__Cone_cm__Matrix_rp.html" title="computes the affine preimage of a cone">affinePreimage(Matrix,Cone,Matrix)</a> -- computes the affine preimage of a cone</span></span></li>
<li><span>affinePreimage(Matrix,Cone), see <span><a href="_affine__Preimage_lp__Matrix_cm__Cone_cm__Matrix_rp.html" title="computes the affine preimage of a cone">affinePreimage(Matrix,Cone,Matrix)</a> -- computes the affine preimage of a cone</span></span></li>
<li><span><a href="_affine__Preimage_lp__Matrix_cm__Cone_cm__Matrix_rp.html" title="computes the affine preimage of a cone">affinePreimage(Matrix,Cone,Matrix)</a> -- computes the affine preimage of a cone</span></li>
<li><span>affinePreimage(Matrix,Polyhedron), see <span><a href="_affine__Preimage_lp__Matrix_cm__Polyhedron_cm__Matrix_rp.html" title="computes the affine preimage of a polyhedron">affinePreimage(Matrix,Polyhedron,Matrix)</a> -- computes the affine preimage of a polyhedron</span></span></li>
<li><span><a href="_affine__Preimage_lp__Matrix_cm__Polyhedron_cm__Matrix_rp.html" title="computes the affine preimage of a polyhedron">affinePreimage(Matrix,Polyhedron,Matrix)</a> -- computes the affine preimage of a polyhedron</span></li>
<li><span>affinePreimage(Polyhedron,Matrix), see <span><a href="_affine__Preimage_lp__Matrix_cm__Polyhedron_cm__Matrix_rp.html" title="computes the affine preimage of a polyhedron">affinePreimage(Matrix,Polyhedron,Matrix)</a> -- computes the affine preimage of a polyhedron</span></span></li>
<li><span>areCompatible(Cone,Cone), see <span><a href="_are__Compatible.html" title="checks if the intersection of two cones is a face of each">areCompatible</a> -- checks if the intersection of two cones is a face of each</span></span></li>
<li><span>cellDecompose(Polyhedron,Matrix), see <span><a href="_cell__Decompose.html" title="computes the regular cell decomposition">cellDecompose</a> -- computes the regular cell decomposition</span></span></li>
<li><span>commonFace(Cone,Cone), see <span><a href="_common__Face.html" title="checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans">commonFace</a> -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans</span></span></li>
<li><span>commonFace(Cone,Fan), see <span><a href="_common__Face.html" title="checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans">commonFace</a> -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans</span></span></li>
<li><span>commonFace(Fan,Cone), see <span><a href="_common__Face.html" title="checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans">commonFace</a> -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans</span></span></li>
<li><span>commonFace(Fan,Fan), see <span><a href="_common__Face.html" title="checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans">commonFace</a> -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans</span></span></li>
<li><span>commonFace(Polyhedron,Polyhedron), see <span><a href="_common__Face.html" title="checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans">commonFace</a> -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans</span></span></li>
<li><span><a href="___Cone_sp_st_sp__Cone.html" title="computes the direct product of two cones">Cone * Cone</a> -- computes the direct product of two cones</span></li>
<li><span><a href="___Cone_sp_st_sp__Polyhedron.html" title="computes the direct product of a cone and a polyhedron">Cone * Polyhedron</a> -- computes the direct product of a cone and a polyhedron</span></li>
<li><span><a href="___Cone_sp_pl_sp__Cone.html" title="computes the Minkowski sum of two cones">Cone + Cone</a> -- computes the Minkowski sum of two cones</span></li>
<li><span><a href="___Cone_sp_pl_sp__Polyhedron.html" title="computes the Minkowski sum of a cone and a polyhedron">Cone + Polyhedron</a> -- computes the Minkowski sum of a cone and a polyhedron</span></li>
<li><span><a href="___Cone_sp_eq_eq_sp__Cone.html" title="equality">Cone == Cone</a> -- equality</span></li>
<li><span><a href="___Cone_sp_qu_sp__Cone.html" title="compares the Cones">Cone ? Cone</a> -- compares the Cones</span></li>
<li><span>cones(ZZ,Fan), see <span><a href="_cones.html" title="computes all cones of a fan of a certain dimension">cones</a> -- computes all cones of a fan of a certain dimension</span></span></li>
<li><span>contains(Cone,Cone), see <span><a href="_contains.html" title="checks if the first argument contains the second argument">contains</a> -- checks if the first argument contains the second argument</span></span></li>
<li><span>contains(Cone,Matrix), see <span><a href="_contains.html" title="checks if the first argument contains the second argument">contains</a> -- checks if the first argument contains the second argument</span></span></li>
<li><span>contains(Cone,Polyhedron), see <span><a href="_contains.html" title="checks if the first argument contains the second argument">contains</a> -- checks if the first argument contains the second argument</span></span></li>
<li><span>contains(Fan,Cone), see <span><a href="_contains.html" title="checks if the first argument contains the second argument">contains</a> -- checks if the first argument contains the second argument</span></span></li>
<li><span>contains(List,Cone), see <span><a href="_contains.html" title="checks if the first argument contains the second argument">contains</a> -- checks if the first argument contains the second argument</span></span></li>
<li><span>contains(List,Polyhedron), see <span><a href="_contains.html" title="checks if the first argument contains the second argument">contains</a> -- checks if the first argument contains the second argument</span></span></li>
<li><span>contains(Polyhedron,Cone), see <span><a href="_contains.html" title="checks if the first argument contains the second argument">contains</a> -- checks if the first argument contains the second argument</span></span></li>
<li><span>contains(Polyhedron,Matrix), see <span><a href="_contains.html" title="checks if the first argument contains the second argument">contains</a> -- checks if the first argument contains the second argument</span></span></li>
<li><span>contains(Polyhedron,Polyhedron), see <span><a href="_contains.html" title="checks if the first argument contains the second argument">contains</a> -- checks if the first argument contains the second argument</span></span></li>
<li><span>convexHull(Polyhedron,Polyhedron), see <span><a href="_convex__Hull.html" title="computing the convex hull of points, rays and polyhedra">convexHull</a> -- computing the convex hull of points, rays and polyhedra</span></span></li>
<li><span><a href="_direct__Product_lp__Cone_cm__Cone_rp.html" title="computes the direct product of polyhedra and cones">directProduct(Cone,Cone)</a> -- computes the direct product of polyhedra and cones</span></li>
<li><span>directProduct(Cone,Polyhedron), see <span><a href="_direct__Product_lp__Cone_cm__Cone_rp.html" title="computes the direct product of polyhedra and cones">directProduct(Cone,Cone)</a> -- computes the direct product of polyhedra and cones</span></span></li>
<li><span>directProduct(Polyhedron,Cone), see <span><a href="_direct__Product_lp__Cone_cm__Cone_rp.html" title="computes the direct product of polyhedra and cones">directProduct(Cone,Cone)</a> -- computes the direct product of polyhedra and cones</span></span></li>
<li><span>directProduct(Polyhedron,Polyhedron), see <span><a href="_direct__Product_lp__Cone_cm__Cone_rp.html" title="computes the direct product of polyhedra and cones">directProduct(Cone,Cone)</a> -- computes the direct product of polyhedra and cones</span></span></li>
<li><span><a href="_direct__Product_lp__Fan_cm__Fan_rp.html" title="computes the direct product of two fans">directProduct(Fan,Fan)</a> -- computes the direct product of two fans</span></li>
<li><span><a href="_dual__Face__Lattice_lp__Z__Z_cm__Cone_rp.html" title="computes the dual face lattice of a cone">dualFaceLattice(ZZ,Cone)</a> -- computes the dual face lattice of a cone</span></li>
<li><span><a href="_dual__Face__Lattice_lp__Z__Z_cm__Polyhedron_rp.html" title="computes the dual face lattice of a polyhedron">dualFaceLattice(ZZ,Polyhedron)</a> -- computes the dual face lattice of a polyhedron</span></li>
<li><span><a href="_face__Lattice_lp__Z__Z_cm__Cone_rp.html" title="computes the face lattice of a cone">faceLattice(ZZ,Cone)</a> -- computes the face lattice of a cone</span></li>
<li><span><a href="_face__Lattice_lp__Z__Z_cm__Polyhedron_rp.html" title="computes the face lattice of a polyhedron">faceLattice(ZZ,Polyhedron)</a> -- computes the face lattice of a polyhedron</span></li>
<li><span>faces(ZZ,Cone), see <span><a href="_faces.html" title="computes all faces of a certain codimension of a Cone or Polyhedron">faces</a> -- computes all faces of a certain codimension of a Cone or Polyhedron</span></span></li>
<li><span>faces(ZZ,Polyhedron), see <span><a href="_faces.html" title="computes all faces of a certain codimension of a Cone or Polyhedron">faces</a> -- computes all faces of a certain codimension of a Cone or Polyhedron</span></span></li>
<li><span><a href="___Fan_sp_st_sp__Fan.html" title="computes the direct product">Fan * Fan</a> -- computes the direct product</span></li>
<li><span><a href="___Fan_sp_eq_eq_sp__Fan.html" title="equality">Fan == Fan</a> -- equality</span></li>
<li><span>imageFan(Matrix,Cone), see <span><a href="_image__Fan.html" title=" computes the fan of the image">imageFan</a> -- computes the fan of the image</span></span></li>
<li><span>incompCones(Cone,Fan), see <span><a href="_incomp__Cones.html" title="returns the pairs of incompatible cones">incompCones</a> -- returns the pairs of incompatible cones</span></span></li>
<li><span>incompCones(Fan,Cone), see <span><a href="_incomp__Cones.html" title="returns the pairs of incompatible cones">incompCones</a> -- returns the pairs of incompatible cones</span></span></li>
<li><span>incompCones(Fan,Fan), see <span><a href="_incomp__Cones.html" title="returns the pairs of incompatible cones">incompCones</a> -- returns the pairs of incompatible cones</span></span></li>
<li><span>inInterior(Matrix,Cone), see <span><a href="_in__Interior.html" title="checks if a point lies in the relative interior of a Cone/Polyhedron">inInterior</a> -- checks if a point lies in the relative interior of a Cone/Polyhedron</span></span></li>
<li><span>inInterior(Matrix,Polyhedron), see <span><a href="_in__Interior.html" title="checks if a point lies in the relative interior of a Cone/Polyhedron">inInterior</a> -- checks if a point lies in the relative interior of a Cone/Polyhedron</span></span></li>
<li><span>intersection(Cone,Cone), see <span><a href="_intersection.html" title="computes the intersection of half-spaces, hyperplanes, cones, and polyhedra">intersection</a> -- computes the intersection of half-spaces, hyperplanes, cones, and polyhedra</span></span></li>
<li><span>intersection(Cone,Polyhedron), see <span><a href="_intersection.html" title="computes the intersection of half-spaces, hyperplanes, cones, and polyhedra">intersection</a> -- computes the intersection of half-spaces, hyperplanes, cones, and polyhedra</span></span></li>
<li><span>intersection(Polyhedron,Cone), see <span><a href="_intersection.html" title="computes the intersection of half-spaces, hyperplanes, cones, and polyhedra">intersection</a> -- computes the intersection of half-spaces, hyperplanes, cones, and polyhedra</span></span></li>
<li><span>intersection(Polyhedron,Polyhedron), see <span><a href="_intersection.html" title="computes the intersection of half-spaces, hyperplanes, cones, and polyhedra">intersection</a> -- computes the intersection of half-spaces, hyperplanes, cones, and polyhedra</span></span></li>
<li><span>isFace(Cone,Cone), see <span><a href="_is__Face.html" title="tests if the first argument is a face of the second">isFace</a> -- tests if the first argument is a face of the second</span></span></li>
<li><span>isFace(Polyhedron,Polyhedron), see <span><a href="_is__Face.html" title="tests if the first argument is a face of the second">isFace</a> -- tests if the first argument is a face of the second</span></span></li>
<li><span>maxFace(Matrix,Cone), see <span><a href="_max__Face.html" title="computes the face of a Polyhedron or Cone where a weight attains its maximum">maxFace</a> -- computes the face of a Polyhedron or Cone where a weight attains its maximum</span></span></li>
<li><span>maxFace(Matrix,Polyhedron), see <span><a href="_max__Face.html" title="computes the face of a Polyhedron or Cone where a weight attains its maximum">maxFace</a> -- computes the face of a Polyhedron or Cone where a weight attains its maximum</span></span></li>
<li><span>minFace(Matrix,Cone), see <span><a href="_min__Face.html" title="computes the face of a Polyhedron or Cone where a weight attains its minimum">minFace</a> -- computes the face of a Polyhedron or Cone where a weight attains its minimum</span></span></li>
<li><span>minFace(Matrix,Polyhedron), see <span><a href="_min__Face.html" title="computes the face of a Polyhedron or Cone where a weight attains its minimum">minFace</a> -- computes the face of a Polyhedron or Cone where a weight attains its minimum</span></span></li>
<li><span>minkowskiSum(Cone,Cone), see <span><a href="_minkowski__Sum.html" title=" computes the Minkowski sum of two convex objects">minkowskiSum</a> -- computes the Minkowski sum of two convex objects</span></span></li>
<li><span>minkowskiSum(Cone,Polyhedron), see <span><a href="_minkowski__Sum.html" title=" computes the Minkowski sum of two convex objects">minkowskiSum</a> -- computes the Minkowski sum of two convex objects</span></span></li>
<li><span>minkowskiSum(Polyhedron,Cone), see <span><a href="_minkowski__Sum.html" title=" computes the Minkowski sum of two convex objects">minkowskiSum</a> -- computes the Minkowski sum of two convex objects</span></span></li>
<li><span>minkowskiSum(Polyhedron,Polyhedron), see <span><a href="_minkowski__Sum.html" title=" computes the Minkowski sum of two convex objects">minkowskiSum</a> -- computes the Minkowski sum of two convex objects</span></span></li>
<li><span><a href="_normal__Cone_lp__Polyhedron_cm__Polyhedron_rp.html" title="computes the normal cone of a face of a polyhedron">normalCone(Polyhedron,Polyhedron)</a> -- computes the normal cone of a face of a polyhedron</span></li>
<li><span>objectiveVector(Polyhedron,Polyhedron), see <span><a href="_objective__Vector.html" title="computes an objective vector of a face of a polyhedron">objectiveVector</a> -- computes an objective vector of a face of a polyhedron</span></span></li>
<li><span><a href="___Polyhedron_sp_st_sp__Cone.html" title="computes the direct product of a polyhedron and a cone">Polyhedron * Cone</a> -- computes the direct product of a polyhedron and a cone</span></li>
<li><span><a href="___Polyhedron_sp_st_sp__Polyhedron.html" title="computes the direct product of two polyhedra">Polyhedron * Polyhedron</a> -- computes the direct product of two polyhedra</span></li>
<li><span><a href="___Polyhedron_sp_pl_sp__Cone.html" title="computes the Minkowski sum of a polyhedron and a cone">Polyhedron + Cone</a> -- computes the Minkowski sum of a polyhedron and a cone</span></li>
<li><span><a href="___Polyhedron_sp_pl_sp__Polyhedron.html" title="computes the Minkowski sum of two polyhedra">Polyhedron + Polyhedron</a> -- computes the Minkowski sum of two polyhedra</span></li>
<li><span><a href="___Polyhedron_sp_eq_eq_sp__Polyhedron.html" title="equality">Polyhedron == Polyhedron</a> -- equality</span></li>
<li><span>posHull(Cone,Cone), see <span><a href="_pos__Hull.html" title="computes the positive hull of rays, cones, and the cone over a polyhedron">posHull</a> -- computes the positive hull of rays, cones, and the cone over a polyhedron</span></span></li>
<li><span>proximum(Matrix,Cone), see <span><a href="_proximum.html" title="computes the proximum of the Polyhedron/Cone to a point in euclidian metric">proximum</a> -- computes the proximum of the Polyhedron/Cone to a point in euclidian metric</span></span></li>
<li><span>proximum(Matrix,Polyhedron), see <span><a href="_proximum.html" title="computes the proximum of the Polyhedron/Cone to a point in euclidian metric">proximum</a> -- computes the proximum of the Polyhedron/Cone to a point in euclidian metric</span></span></li>
<li><span><a href="___Q__Q_sp_st_sp__Polyhedron.html" title="rescales a polyhedron by a given positive factor">QQ * Polyhedron</a> -- rescales a polyhedron by a given positive factor</span></li>
<li><span>ZZ * Polyhedron, see <span><a href="___Q__Q_sp_st_sp__Polyhedron.html" title="rescales a polyhedron by a given positive factor">QQ * Polyhedron</a> -- rescales a polyhedron by a given positive factor</span></span></li>
<li><span>skeleton(ZZ,Fan), see <span><a href="_skeleton.html" title="computes the k-skeleton of a Fan">skeleton</a> -- computes the k-skeleton of a Fan</span></span></li>
<li><span>smallestFace(Matrix,Cone), see <span><a href="_smallest__Face.html" title="determines the smallest face of the Cone/Polyhedron containing a point">smallestFace</a> -- determines the smallest face of the Cone/Polyhedron containing a point</span></span></li>
<li><span>smallestFace(Matrix,Polyhedron), see <span><a href="_smallest__Face.html" title="determines the smallest face of the Cone/Polyhedron containing a point">smallestFace</a> -- determines the smallest face of the Cone/Polyhedron containing a point</span></span></li>
<li><span>stellarSubdivision(Fan,Matrix), see <span><a href="_stellar__Subdivision.html" title="computes the stellar subdivision of the fan by a ray">stellarSubdivision</a> -- computes the stellar subdivision of the fan by a ray</span></span></li>
</ul>
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