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<head><title>Polyhedron -- the class of all convex polyhedra</title>
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<div><h1>Polyhedron -- the class of all convex polyhedra</h1>
<div class="single"><h2>Description</h2>
<div>A Polyhedron represents a rational polyhedron. It can be bounded or unbounded,
need not be full dimensional or may contain a proper affine subspace. It can
be empty or zero dimensional. It is saved as a hash table which contains
the vertices, generating rays, and the basis of the lineality space of the
Polyhedron as well as the defining affine half-spaces and hyperplanes. The
output of a Polyhedron looks like this:<table class="examples"><tr><td><pre>i1 : convexHull(matrix {{0,0,-1,-1},{2,-2,1,-1},{0,0,0,0}},matrix {{1},{0},{0}})
o1 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 5
number of rays => 1
number of vertices => 4
o1 : Polyhedron</pre>
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<p/>
This table displays a short summary of the properties of the Polyhedron.
Note that the number of rays and vertices are modulo the lineality space. So
for example a line in QQ^2 has one vertex and no rays. However, one can not
access the above information directly, because this is just a virtual hash table
generated for the output. The data defining a Polyhedron is extracted
by the functions included in this package. A Polyhedron can be constructed as
the convex hull (<a href="_convex__Hull.html" title="computing the convex hull of points, rays and polyhedra">convexHull</a>) of a set of points and a set of rays or as the
intersection (<a href="_intersection.html" title="computes the intersection of half-spaces, hyperplanes, cones, and polyhedra">intersection</a>) of a set of affine half-spaces and affine hyperplanes.<p/>
For example, consider the square and the square with an emerging ray
for the convex hull:<table class="examples"><tr><td><pre>i2 : V = matrix {{1,1,-1,-1},{1,-1,1,-1}}
o2 = | 1 1 -1 -1 |
| 1 -1 1 -1 |
2 4
o2 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i3 : convexHull V
o3 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 4
number of rays => 0
number of vertices => 4
o3 : Polyhedron</pre>
</td></tr>
<tr><td><pre>i4 : R = matrix {{1},{1}}
o4 = | 1 |
| 1 |
2 1
o4 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i5 : convexHull(V,R)
o5 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 4
number of rays => 1
number of vertices => 3
o5 : Polyhedron</pre>
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<p/>
If we take the intersection of the half-spaces defined by the directions of the
vertices and 1 we get the crosspolytope:<table class="examples"><tr><td><pre>i6 : HS = transpose V
o6 = | 1 1 |
| 1 -1 |
| -1 1 |
| -1 -1 |
4 2
o6 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i7 : v = R || R
o7 = | 1 |
| 1 |
| 1 |
| 1 |
4 1
o7 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i8 : P = intersection(HS,v)
o8 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 4
number of rays => 0
number of vertices => 4
o8 : Polyhedron</pre>
</td></tr>
<tr><td><pre>i9 : vertices P
o9 = | -1 1 0 0 |
| 0 0 -1 1 |
2 4
o9 : Matrix QQ <--- QQ</pre>
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<p/>
This can for example be embedded in 3-space on height 1:<table class="examples"><tr><td><pre>i10 : HS = HS | matrix {{0},{0},{0},{0}}
o10 = | 1 1 0 |
| 1 -1 0 |
| -1 1 0 |
| -1 -1 0 |
4 3
o10 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i11 : HP = matrix {{0,0,1}}
o11 = | 0 0 1 |
1 3
o11 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i12 : w = matrix {{1}}
o12 = | 1 |
1 1
o12 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i13 : P = intersection(HS,v,HP,w)
o13 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 4
number of rays => 0
number of vertices => 4
o13 : Polyhedron</pre>
</td></tr>
<tr><td><pre>i14 : vertices P
o14 = | -1 1 0 0 |
| 0 0 -1 1 |
| 1 1 1 1 |
3 4
o14 : Matrix QQ <--- QQ</pre>
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</table>
<p/>
See also <a href="___Working_spwith_sppolyhedra.html" title="">Working with polyhedra</a>.</div>
</div>
<div class="waystouse"><h2>Functions and methods returning a convex polyhedron :</h2>
<ul><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="_bipyramid.html" title="computes the bipyramid over a polyhedron">bipyramid</a> -- computes the bipyramid over a polyhedron</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="_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="_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="_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="_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="_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="_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="_pyramid.html" title="computes the pyramid over a polyhedron">pyramid</a> -- computes the pyramid over a polyhedron</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="_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>
</ul>
<h2>Methods that use a convex polyhedron :</h2>
<ul><li><span>affineHull(Polyhedron), see <span><a href="_affine__Hull.html" title="computes the affine hull of a polyhedron">affineHull</a> -- computes the affine hull of a polyhedron</span></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(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>ambDim(Polyhedron), see <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></span></li>
<li><span>bipyramid(Polyhedron), see <span><a href="_bipyramid.html" title="computes the bipyramid over a polyhedron">bipyramid</a> -- computes the bipyramid over a polyhedron</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(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__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__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>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(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="_dim_lp__Polyhedron_rp.html" title="computes the dimension of a polyhedron">dim(Polyhedron)</a> -- computes the dimension of a polyhedron</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>dualFaceLattice(Polyhedron), see <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></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>ehrhart(Polyhedron), see <span><a href="_ehrhart.html" title="calculates the Ehrhart polynomial of a polytope">ehrhart</a> -- calculates the Ehrhart polynomial of a polytope</span></span></li>
<li><span>faceFan(Polyhedron), see <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></span></li>
<li><span>faceLattice(Polyhedron), see <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></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,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>fVector(Polyhedron), see <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></span></li>
<li><span>halfspaces(Polyhedron), see <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></span></li>
<li><span>hyperplanes(Polyhedron), see <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></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>interiorPoint(Polyhedron), see <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></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>isCompact(Polyhedron), see <span><a href="_is__Compact.html" title="checks compactness of a Polyhedron">isCompact</a> -- checks compactness of a Polyhedron</span></span></li>
<li><span>isEmpty(Polyhedron), see <span><a href="_is__Empty.html" title="checks if a Polyhedron is empty">isEmpty</a> -- checks if a Polyhedron is empty</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><a href="_is__Normal_lp__Polyhedron_rp.html" title="checks if a polytope is normal in the ambient lattice">isNormal(Polyhedron)</a> -- checks if a polytope is normal in the ambient lattice</span></li>
<li><span>latticePoints(Polyhedron), see <span><a href="_lattice__Points.html" title="computes the lattice points of a polytope">latticePoints</a> -- computes the lattice points of a polytope</span></span></li>
<li><span>linSpace(Polyhedron), see <span><a href="_lin__Space.html" title="computes a basis of the lineality space">linSpace</a> -- computes a basis of the lineality space</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,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,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>minkSummandCone(Polyhedron), see <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></span></li>
<li><span><a href="_net_lp__Polyhedron_rp.html" title="displays characteristics of a polyhedron">net(Polyhedron)</a> -- displays characteristics of a polyhedron</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>normalFan(Polyhedron), see <span><a href="_normal__Fan.html" title="computes the normalFan of a polyhedron">normalFan</a> -- computes the normalFan of a polyhedron</span></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>polar(Polyhedron), see <span><a href="_polar.html" title=" computes the polar of a polyhedron">polar</a> -- computes the polar 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(Polyhedron), 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,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>pyramid(Polyhedron), see <span><a href="_pyramid.html" title="computes the pyramid over a polyhedron">pyramid</a> -- computes the pyramid over a polyhedron</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>rays(Polyhedron), see <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></span></li>
<li><span>secondaryPolytope(Polyhedron), see <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></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>sublatticeBasis(Polyhedron), see <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></span></li>
<li><span>tailCone(Polyhedron), see <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></span></li>
<li><span>toSublattice(Polyhedron), see <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></span></li>
<li><span>triangulate(Polyhedron), see <span><a href="_triangulate.html" title="computes a triangulation of a polytope">triangulate</a> -- computes a triangulation of a polytope</span></span></li>
<li><span>vertexEdgeMatrix(Polyhedron), see <span><a href="_vertex__Edge__Matrix.html" title="computes the vertex-edge-relations matrix">vertexEdgeMatrix</a> -- computes the vertex-edge-relations matrix</span></span></li>
<li><span>vertexFacetMatrix(Polyhedron), see <span><a href="_vertex__Facet__Matrix.html" title="computes the vertex-facet-relations matrix">vertexFacetMatrix</a> -- computes the vertex-facet-relations matrix</span></span></li>
<li><span>vertices(Polyhedron), see <span><a href="_vertices.html" title="displays the vertices of a Polyhedron">vertices</a> -- displays the vertices of a Polyhedron</span></span></li>
<li><span>volume(Polyhedron), see <span><a href="_volume.html" title="computes the volume of a polytope">volume</a> -- computes the volume of a polytope</span></span></li>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Polyhedron.html" title="the class of all convex polyhedra">Polyhedron</a> is <span>a <a href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a href="___Polyhedral__Object.html" title="the class of all polyhedral objects in Polyhedra">PolyhedralObject</a> < <a href="../../Macaulay2Doc/html/___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="../../Macaulay2Doc/html/___Thing.html" title="the class of all things">Thing</a>.</p>
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