GAP (Convex) - Chapter 6: Polytope

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6 Polytope
6.1 Polytope: Category and Representations
6.1-1 IsPolytope
6.2 Polytope: Properties
6.2-1 IsNotEmpty
6.2-2 IsLatticePolytope
6.2-3 IsVeryAmple
6.2-4 IsNormalPolytope
6.2-5 IsSimplicial
6.2-6 IsSimplePolytope
6.3 Polytope: Attributes
6.3-1 Vertices
6.3-2 LatticePoints
6.3-3 FacetInequalities
6.3-4 VerticesInFacets
6.3-5 AffineCone
6.3-6 NormalFan
6.3-7 RelativeInteriorLatticePoints
6.4 Polytope: Methods
6.4-1 *
6.4-2 #
6.5 Polytope: Constructors
6.5-1 Polytope
6.5-2 PolytopeByInequalities
6.6 Polytope: Examples
6.6-1 Polytope example

### 6 Polytope

#### 6.1 Polytope: Category and Representations

##### 6.1-1 IsPolytope

`‣ IsPolytope`( M ) | ( category )

Returns: `true` or `false`

The GAP category of a polytope. Every polytope is a convex object.

Remember: Every cone is a convex object.

#### 6.2 Polytope: Properties

##### 6.2-1 IsNotEmpty

`‣ IsNotEmpty`( poly ) | ( property )

Returns: `true` or `false`

Checks if the polytope poly is not empty.

##### 6.2-2 IsLatticePolytope

`‣ IsLatticePolytope`( poly ) | ( property )

Returns: `true` or `false`

Checks if the polytope poly is a lattice polytope, i.e. all its vertices are lattice points.

##### 6.2-3 IsVeryAmple

`‣ IsVeryAmple`( poly ) | ( property )

Returns: `true` or `false`

Checks if the polytope poly is very ample.

##### 6.2-4 IsNormalPolytope

`‣ IsNormalPolytope`( poly ) | ( property )

Returns: `true` or `false`

Checks if the polytope poly is normal.

##### 6.2-5 IsSimplicial

`‣ IsSimplicial`( poly ) | ( property )

Returns: `true` or `false`

Checks if the polytope poly is simplicial.

##### 6.2-6 IsSimplePolytope

`‣ IsSimplePolytope`( poly ) | ( property )

Returns: `true` or `false`

Checks if the polytope poly is simple.

#### 6.3 Polytope: Attributes

##### 6.3-1 Vertices

`‣ Vertices`( poly ) | ( attribute )

Returns: a list

Returns the vertices of the polytope poly. For reasons, the corresponding tester is HasVerticesOfPolytopes

##### 6.3-2 LatticePoints

`‣ LatticePoints`( poly ) | ( attribute )

Returns: a list

Returns the lattice points of the polytope poly.

##### 6.3-3 FacetInequalities

`‣ FacetInequalities`( poly ) | ( attribute )

Returns: a list

Returns the facet inequalities for the polytope poly.

##### 6.3-4 VerticesInFacets

`‣ VerticesInFacets`( poly ) | ( attribute )

Returns: a list

Returns the incidence matrix of vertices and facets of the polytope poly.

##### 6.3-5 AffineCone

`‣ AffineCone`( poly ) | ( attribute )

Returns: a cone

Returns the affine cone of the polytope poly.

##### 6.3-6 NormalFan

`‣ NormalFan`( poly ) | ( attribute )

Returns: a fan

Returns the normal fan of the polytope poly.

##### 6.3-7 RelativeInteriorLatticePoints

`‣ RelativeInteriorLatticePoints`( poly ) | ( attribute )

Returns: a list

Returns the lattice points in the relative interior of the polytope poly.

#### 6.4 Polytope: Methods

##### 6.4-1 *

`‣ *`( polytope1, polytope2 ) | ( operation )

Returns: a polytope

Returns the Cartesian product of the polytopes polytope1 and polytope2.

##### 6.4-2 #

`‣ #`( polytope1, polytope2 ) | ( operation )

Returns: a polytope

Returns the Minkowski sum of the polytopes polytope1 and polytope2.

#### 6.5 Polytope: Constructors

##### 6.5-1 Polytope

`‣ Polytope`( points ) | ( operation )

Returns: a polytope

Returns a polytope that is the convex hull of the points points.

##### 6.5-2 PolytopeByInequalities

`‣ PolytopeByInequalities`( ineqs ) | ( operation )

Returns: a polytope

Returns a polytope defined by the inequalities ineqs.

#### 6.6 Polytope: Examples

##### 6.6-1 Polytope example

    gap> P := Polytope( [ [ 2, 0 ], [ 0, 2 ], [ -1, -1 ] ] );
    <A polytope in |R^2>
    gap> IsVeryAmple( P );
    true
    gap> LatticePoints( P );
    [ [ -1, -1 ], [ 0, 0 ], [ 0, 1 ],
    [ 0, 2 ], [ 1, 0 ], [ 1, 1 ], [ 2, 0 ] ]
    gap> NFP := NormalFan( P );
    <A complete fan in |R^2>
    gap> C1 := MaximalCones( NFP )[ 1 ];
    <A cone in |R^2>
    gap> RayGenerators( C1 );
    [ [ -1, -1 ], [ -1, 3 ] ]
    gap> IsRegularFan( NFP );
    true

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