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### 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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