# Base class for polyhedra over $$\ZZ$$¶

class sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ(parent, Vrep, Hrep, **kwds)

Base class for Polyhedra over $$\ZZ$$

TESTS:

sage: p = Polyhedron([(0,0)], base_ring=ZZ);  p
A 0-dimensional polyhedron in ZZ^2 defined as the convex hull of 1 vertex
sage: TestSuite(p).run(skip='_test_pickling')
Minkowski_decompositions()

Return all Minkowski sums that add up to the polyhedron.

OUTPUT:

A tuple consisting of pairs $$(X,Y)$$ of $$\ZZ$$-polyhedra that add up to self. All pairs up to exchange of the summands are returned, that is, $$(Y,X)$$ is not included if $$(X,Y)$$ already is.

EXAMPLES:

sage: square = Polyhedron(vertices=[(0,0),(1,0),(0,1),(1,1)])
sage: square.Minkowski_decompositions()
((A 0-dimensional polyhedron in ZZ^2 defined as the convex hull of 1 vertex,
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices),
(A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices,
A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices))

Example from http://cgi.di.uoa.gr/~amantzaf/geo/

sage: Q = Polyhedron(vertices=[(4,0), (6,0), (0,3), (4,3)])
sage: R = Polyhedron(vertices=[(0,0), (5,0), (8,4), (3,2)])
sage: (Q+R).Minkowski_decompositions()
((A 0-dimensional polyhedron in ZZ^2 defined as the convex hull of 1 vertex,
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 7 vertices),
(A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices,
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices),
(A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices,
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 7 vertices),
(A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 5 vertices,
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices),
(A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices,
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 7 vertices),
(A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 5 vertices,
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices),
(A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices,
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 7 vertices),
(A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices,
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 6 vertices))

sage: [ len(square.dilation(i).Minkowski_decompositions())
...     for i in range(6) ]
[1, 2, 5, 8, 13, 18]
sage: [ ceil((i^2+2*i-1)/2)+1 for i in range(10) ]
[1, 2, 5, 8, 13, 18, 25, 32, 41, 50]
fibration_generator(dim)

Generate the lattice polytope fibrations.

For the purposes of this function, a lattice polytope fiber is a sub-lattice polytope. Projecting the plane spanned by the subpolytope to a point yields another lattice polytope, the base of the fibration.

INPUT:

• dim – integer. The dimension of the lattice polytope fiber.

OUTPUT:

A generator yielding the distinct lattice polytope fibers of given dimension.

EXAMPLES:

sage: P = Polyhedron(toric_varieties.P4_11169().fan().rays(), base_ring=ZZ)
sage: list( P.fibration_generator(2) )
[A 2-dimensional polyhedron in ZZ^4 defined as the convex hull of 3 vertices]
find_translation(translated_polyhedron)

Return the translation vector to translated_polyhedron.

INPUT:

• translated_polyhedron – a polyhedron.

OUTPUT:

A $$\ZZ$$-vector that translates self to translated_polyhedron. A ValueError is raised if translated_polyhedron is not a translation of self, this can be used to check that two polyhedra are not translates of each other.

EXAMPLES:

sage: X = polytopes.n_cube(3)
sage: X.find_translation(X + vector([2,3,5]))
(2, 3, 5)
sage: X.find_translation(2*X)
Traceback (most recent call last):
...
ValueError: polyhedron is not a translation of self
has_IP_property()

Test whether the polyhedron has the IP property.

The IP (interior point) property means that

• self is compact (a polytope).
• self contains the origin as an interior point.

This implies that

• self is full-dimensional.
• The dual polyhedron is again a polytope (that is, a compact polyhedron), though not necessarily a lattice polytope.

EXAMPLES:

sage: Polyhedron([(1,1),(1,0),(0,1)], base_ring=ZZ).has_IP_property()
False
sage: Polyhedron([(0,0),(1,0),(0,1)], base_ring=ZZ).has_IP_property()
False
sage: Polyhedron([(-1,-1),(1,0),(0,1)], base_ring=ZZ).has_IP_property()
True

REFERENCES:

 [PALP] Maximilian Kreuzer, Harald Skarke: “PALP: A Package for Analyzing Lattice Polytopes with Applications to Toric Geometry” Comput.Phys.Commun. 157 (2004) 87-106 Arxiv math/0204356
is_lattice_polytope()

Return whether the polyhedron is a lattice polytope.

OUTPUT:

True if the polyhedron is compact and has only integral vertices, False otherwise.

EXAMPLES:

sage: polytopes.cross_polytope(3).is_lattice_polytope()
True
sage: polytopes.regular_polygon(5).is_lattice_polytope()
False
is_reflexive()

EXAMPLES:

sage: p = Polyhedron(vertices=[(1,0,0),(0,1,0),(0,0,1),(-1,-1,-1)], base_ring=ZZ)
sage: p.is_reflexive()
True
polar()

Return the polar (dual) polytope.

The polytope must have the IP-property (see has_IP_property()), that is, the origin must be an interior point. In particular, it must be full-dimensional.

OUTPUT:

The polytope whose vertices are the coefficient vectors of the inequalities of self with inhomogeneous term normalized to unity.

EXAMPLES:

sage: p = Polyhedron(vertices=[(1,0,0),(0,1,0),(0,0,1),(-1,-1,-1)], base_ring=ZZ)
sage: p.polar()
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 4 vertices
sage: type(_)
<class 'sage.geometry.polyhedron.backend_ppl.Polyhedra_ZZ_ppl_with_category.element_class'>
sage: p.polar().base_ring()
Integer Ring

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