Quelle _Chapter_Polytopes.xml Sprache: XML

<?xml version="1.0" encoding="UTF-8"?>


<Chapter Label="Chapter_Polytopes">
<Heading>Polytopes</Heading>

<Section Label="Chapter_Polytopes_Section_Creating_polytopes">
<Heading>Creating polytopes</Heading>

<ManSection>
<Oper Arg="L" Name="PolytopeByInequalities" Label="for IsList"/>
<Returns>a Polytope Object
</Returns>
<Description>
The operation takes a list <Math>L</Math> of lists <Math>[L_1, L_2, ...]</Math> where each <Math>L_j</Math> represents
an inequality and returns the polytope defined by them (if they define a polytope).
For example the <Math>j</Math>'th entry

L_j = [c_j,a_{j1},a_{j2},...,a_{jn}]

corresponds to the inequality
<Math>c_j+\sum_{i=1}^n a_{ji}x_i \geq 0</Math>.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="L" Name="Polytope" Label="for IsList"/>
<Returns>a Polytope Object
</Returns>
<Description>
The operation takes the list of the vertices and returns the polytope defined by them.
</Description>
</ManSection>

</Section>

<Section Label="Chapter_Polytopes_Section_Attributes">
<Heading>Attributes</Heading>

<ManSection>
  <Attr Arg="P" Name="ExternalCddPolytope" Label="for IsPolytope"/>
<Returns>a CddPolyhedron
</Returns>
<Description>
Converts the polytope to a CddPolyhedron. The operations of CddInterface can then be applied
on this polyhedron.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="P" Name="LatticePoints" Label="for IsPolytope"/>
<Returns>a List
</Returns>
<Description>
The operation returns the list of integer points inside the polytope.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="P" Name="RelativeInteriorLatticePoints" Label="for IsPolytope"/>
<Returns>a List
</Returns>
<Description>
The operation returns the interior lattice points inside the polytope.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="P" Name="VerticesOfPolytope" Label="for IsPolytope"/>
<Returns>a list of lists
</Returns>
<Description>
The operation returns the vertices of the polytope
</Description>
</ManSection>

<ManSection>
  <Oper Arg="P" Name="Vertices" Label="for IsPolytope"/>
<Returns>a list of lists
</Returns>
<Description>
The same output as <C>VerticesOfPolytope</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="P" Name="DefiningInequalities" Label="for IsPolytope"/>
<Returns>a list of lists
</Returns>
<Description>
The operation returns the defining inequalities of the polytope.
I.e., a list of lists <Math>[L_1, L_2, ...]</Math> where each
<Math>L_j=[c_j,a_{j1},a_{j2},...,a_{jn}]</Math> represents the inequality
<Math>c_j+\sum_{i=1}^n a_{ji}x_i \geq 0</Math>. If <Math>L</Math> and <Math>-L</Math> occur in the
output then <Math>L</Math> is called a defining-equality of the polytope.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="P" Name="EqualitiesOfPolytope" Label="for IsPolytope"/>
<Returns>a list of lists
</Returns>
<Description>
The operation returns the defining-equalities of the polytope.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="P" Name="FacetInequalities" Label="for IsPolytope"/>
<Returns>a list of lists
</Returns>
<Description>
The operation returns the list of the inequalities of the facets.
Each defining inequality that is not defining-equality of the
polytope is a facet inequality.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="P" Name="VerticesInFacets" Label="for IsPolytope"/>
<Returns>a list of lists
</Returns>
<Description>
The operation returns list of lists <Math>L</Math>. The entries of each <Math>L_j</Math>
in <Math>L</Math> consists of <Math>0</Math>'s or

1

's. For instance, if <Math>L_j=[1,0,0,1,0,1]</Math>, then
The polytope has <Math>6</Math> vertices and the vertices of the <Math>j</Math>'th facet are

\{V_1,V_4,V_6\}

.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="P" Name="NormalFan" Label="for IsPolytope"/>
<Returns>a fan
</Returns>
<Description>
The operation returns the normal fan of the given polytope.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="P" Name="FaceFan" Label="for IsPolytope"/>
<Returns>a fan
</Returns>
<Description>
The operation returns the face fan of the given polytope. Remember that the face fan of a polytope is isomorphic to the normal fan of its
polar polytope.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="P" Name="AffineCone" Label="for IsPolytope"/>
<Returns>a cone
</Returns>
<Description>
If the ambient space of the polytope is <Math>\mathrm{R}^n</Math>, then the output is a cone in
<Math>\mathrm{R}^{n+1}</Math>. The defining rays of the cone are
<Math>{[a_{j1},a_{j2},...,a_{jn},1]}_j</Math> such that <Math>V_j=[a_{j1},a_{j2},...,a_{jn}]</Math> is
a vertex in the polytope.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="P" Name="PolarPolytope" Label="for IsPolytope"/>
<Returns>a Polytope
</Returns>
<Description>
The operation returns the polar polytope of the given polytope.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="P" Name="DualPolytope" Label="for IsPolytope"/>
<Returns>a Polytope
</Returns>
<Description>
The operation returns the dual polytope of the given polytope.
</Description>
</ManSection>

</Section>

<Section Label="Chapter_Polytopes_Section_Properties">
<Heading>Properties</Heading>

<ManSection>
  <Prop Arg="P" Name="IsEmpty" Label="for IsPolytope"/>
<Returns>a true or false
</Returns>
<Description>
Returns whether the polytope empty or not
</Description>
</ManSection>

<ManSection>
  <Prop Arg="P" Name="IsLatticePolytope" Label="for IsPolytope"/>
<Returns>a true or false
</Returns>
<Description>
Returns whether the polytope is lattice polytope or not.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="P" Name="IsVeryAmple" Label="for IsPolytope"/>
<Returns>a true or false
</Returns>
<Description>
Returns whether the polytope is very ample or not.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="P" Name="IsNormalPolytope" Label="for IsPolytope"/>
<Returns>a true or false
</Returns>
<Description>
Returns whether the polytope is normal or not.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="P" Name="IsSimplicial" Label="for IsPolytope"/>
<Returns>a true or false
</Returns>
<Description>
Returns whether the polytope is simplicial or not.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="P" Name="IsSimplexPolytope" Label="for IsPolytope"/>
<Returns>a true or false
</Returns>
<Description>
Returns whether the polytope is simplex polytope or not.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="P" Name="IsSimplePolytope" Label="for IsPolytope"/>
<Returns>a true or false
</Returns>
<Description>
Returns whether the polytope is simple or not.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="P" Name="IsReflexive" Label="for IsPolytope"/>
<Returns>a true or false
</Returns>
<Description>
Returns whether the polytope is reflexive or not, i.e., if its dual polytope is lattice
polytope.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="P" Name="IsFanoPolytope" Label="for IsPolytope"/>
<Returns>a true or false
</Returns>
<Description>
returns whether the polytope is Fano or not. Fano polytope is a full dimensional lattice polytope whose vertices are
primitive elements in the containing lattice, i.e., each vertex is not a positive integer multiple of any other lattice element.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="P" Name="IsCanonicalFanoPolytope" Label="for IsPolytope"/>
<Returns>a true or false
</Returns>
<Description>
returns whether the polytope is canonical Fano or not. A canonical Fano polytope is a full dimensional lattice polytope whose relative
interior contains only one lattice point, namely the origin.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="P" Name="IsTerminalFanoPolytope" Label="for IsPolytope"/>
<Returns>a true or false
</Returns>
<Description>
returns whether the polytope is terminal Fano or not. A terminal Fano polytope is a full dimensional lattice polytope whose
lattice points are its vertices and the origin.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="P" Name="IsSmoothFanoPolytope" Label="for IsPolytope"/>
<Returns>a true or false
</Returns>
<Description>
Returns whether the polytope is smooth fano polytope or not, i.e, if the vertices in each facet form a basis for the containing lattice or not.
polytope.
</Description>
</ManSection>

</Section>

<Section Label="Chapter_Polytopes_Section_Operations_on_polytopes">
<Heading>Operations on polytopes</Heading>

<ManSection>
  <Oper Arg="P1, P2" Name="\+" Label="for IsPolytope, IsPolytope"/>
<Returns>a polytope
</Returns>
<Description>
The output is Minkowski sum of the input polytopes.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="n, P" Name="\*" Label="for IsInt, IsPolytope"/>
<Returns>a polytope
</Returns>
<Description>
The output is Minkowski sum of the input polytope with itself <Math>n</Math> times.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="P1, P2" Name="IntersectionOfPolytopes" Label="for IsPolytope, IsPolytope"/>
<Returns>a polytope
</Returns>
<Description>
The output is the intersection of the input polytopes.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="P" Name="RandomInteriorPoint" Label="for IsPolytope"/>
<Returns>a list
</Returns>
<Description>
Returns a random interior point in the polytope.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="M, P" Name="IsInteriorPoint" Label="for IsList,IsPolytope"/>
<Returns>true or false
</Returns>
<Description>
Checks if the given point is interior point of the polytope.
</Description>
</ManSection>

<#Include Label="example2">

</Section>

</Chapter>

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