Polyhedron partition¶
Polyhedron partitions
EXAMPLES:
A polyhedron partition:
sage: from slabbe import PolyhedronPartition
sage: h = 1/3
sage: p = Polyhedron([(0,h),(0,1),(h,1)])
sage: q = Polyhedron([(0,0), (0,h), (h,1), (h,0)])
sage: r = Polyhedron([(h,1), (1,1), (1,h), (h,0)])
sage: s = Polyhedron([(h,0), (1,0), (1,h)])
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s})
sage: P.is_pairwise_disjoint()
True
sage: list(P)
[(0, A 2dimensional polyhedron in QQ^2 defined as the convex hull of 3 vertices),
(1, A 2dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices),
(2, A 2dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices),
(3, A 2dimensional polyhedron in QQ^2 defined as the convex hull of 3 vertices)]
sage: G = P.plot()
AUTHORS:
Sébastien Labbé, November 2017, initial version of polyhedron partitions

class
slabbe.polyhedron_partition.
PolyhedronPartition
(atoms, base_ring=None)¶ Bases:
object
Return a partition into polyhedron.
Note: Many atoms may share the same key.
INPUT:
atoms
– list of polyhedron or dict of key > polyhedron or list of (key, polyhedron)base_ring
– base ring (default:None
) of the vertices
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: P Polyhedron partition of 3 atoms with 3 letters
sage: P.is_pairwise_disjoint() True sage: P.volume() 1 sage: G = P.plot()
From a dict:
sage: PolyhedronPartition(dict(a=p,b=q,c=r)) Polyhedron partition of 3 atoms with 3 letters
From a list of (key, polyhedron):
sage: PolyhedronPartition([(9,p),(8,q),(9,r)]) Polyhedron partition of 3 atoms with 2 letters

alphabet
()¶ EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([(3,p), (5,q), (9,r)]) sage: P.alphabet() {3, 5, 9} sage: P = PolyhedronPartition([(3,p), (5,q), (3,r)]) sage: P.alphabet() {3, 5}

alphabet_size
()¶ EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([(3,p), (5,q), (9,r)]) sage: P.alphabet_size() 3 sage: P = PolyhedronPartition([(3,p), (5,q), (3,r)]) sage: P.alphabet_size() 2

ambient_space
()¶ EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: P.ambient_space() Vector space of dimension 2 over Rational Field

apply_linear_map
(M)¶ Apply a matrix to every polyhedron of the partition.
INPUT:
M
– a matrix
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/3 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (h,0)]) sage: r = Polyhedron([(h,1), (1,1), (1,h), (h,0)]) sage: s = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s})
Vertical symmetry:
sage: M = diagonal_matrix((1,1)) sage: P = P.apply_linear_map(M) sage: P = P.translate((1,0)) sage: P Polyhedron partition of 4 atoms with 4 letters

atoms
()¶ EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: P.atoms() [A 2dimensional polyhedron in QQ^2 defined as the convex hull of 3 vertices, A 2dimensional polyhedron in QQ^2 defined as the convex hull of 6 vertices, A 2dimensional polyhedron in QQ^2 defined as the convex hull of 3 vertices]

base_ring
()¶ EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: P.base_ring() Rational Field

cached_atoms_set
()¶ EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: P.cached_atoms_set() {A 2dimensional polyhedron in QQ^2 defined as the convex hull of ... vertices, A 2dimensional polyhedron in QQ^2 defined as the convex hull of ... vertices, A 2dimensional polyhedron in QQ^2 defined as the convex hull of ... vertices}

code
(p)¶ Returns in which atom the polyhedron lives in.
INPUT:
p
– a polyhedron
OUTPUT:
integer (for the ith atom)
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/3 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (h,0)]) sage: r = Polyhedron([(h,1), (1,1), (1,h), (h,0)]) sage: s = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s}) sage: P.code(p) 0 sage: P.code(q) 1 sage: t = Polyhedron([(0, 8/9), (0, 1), (1/9, 1)]) sage: P.code(t) 0
TESTS:
sage: t = Polyhedron([(0, 1/9), (0, 1), (1/9, 1)]) sage: P.code(t) Traceback (most recent call last): ... ValueError: polyhedron p whose vertices are (A vertex at (0, 1), A vertex at (0, 1/9), A vertex at (1/9, 1)) lies in no atom

domain
()¶ Return the domain of the partition.
OUTPUT:
a polyhedron
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/3 sage: p = Polyhedron([(0,0),(h,0),(h,1),(0,1)]) sage: q = Polyhedron([(1,0),(h,0),(h,1),(1,1)]) sage: P = PolyhedronPartition({0:p, 1:q}) sage: P.domain() A 2dimensional polyhedron in QQ^2 defined as the convex hull of 4 vertices sage: P.domain().vertices() (A vertex at (0, 0), A vertex at (0, 1), A vertex at (1, 0), A vertex at (1, 1))

edges
()¶ Return the edges of partition (one copy of each edge).
Note
If there are vertices of atoms on the interior of the edge of another atom, then, the overlapping edges will be repeated.
OUTPUT:
set of sorted pair of immutable vectors
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: sorted(P.edges()) [((0, 0), (0, 1/2)), ((0, 0), (1/2, 0)), ((0, 1/2), (0, 1)), ((0, 1/2), (1/2, 1)), ((0, 1), (1/2, 1)), ((1/2, 0), (1, 0)), ((1/2, 0), (1, 1/2)), ((1/2, 1), (1, 1)), ((1, 0), (1, 1/2)), ((1, 1/2), (1, 1))]
Irrational partition:
sage: z = polygen(QQ, 'z') #z = QQ['z'].0 # same as sage: K = NumberField(z**2z1, 'phi', embedding=RR(1.6)) sage: phi = K.gen() sage: h = 1/phi^2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (h,0)]) sage: r = Polyhedron([(h,1), (1,1), (1,h), (h,0)]) sage: s = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s}, base_ring=K) sage: sorted(P.edges()) [((0, 0), (0, phi + 2)), ((0, 0), (phi + 2, 0)), ((0, phi + 2), (0, 1)), ((0, phi + 2), (phi + 2, 1)), ((0, 1), (phi + 2, 1)), ((phi + 2, 0), (phi + 2, 1)), ((phi + 2, 0), (1, 0)), ((phi + 2, 0), (1, phi + 2)), ((phi + 2, 1), (1, 1)), ((1, 0), (1, phi + 2)), ((1, phi + 2), (1, 1))]

is_equal_up_to_relabeling
(other)¶ Return whether two partitions are equivalent, i.e., are equal up to a permutation of the coding.
The coding is not considered.
INPUT:
other
– a partition
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: Q = PolyhedronPartition([p,q]) sage: R = PolyhedronPartition({'asd':q, 'yo':r, 'foo':p}) sage: P.is_equal_up_to_relabeling(P) True sage: P.is_equal_up_to_relabeling(Q) False sage: P.is_equal_up_to_relabeling(R) True

is_pairwise_disjoint
()¶ Return whether atoms of the partition are pairwise disjoint.
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: P.is_pairwise_disjoint() True

keys_permutation
(other)¶ Return a relabelling permutation of the keys for self to look like other.
Note
currently, the code works only if the coding of self and other is injective, i.e., no two polyhedron are coded by the same letter.
INPUT:
other
– a polyhedron partition (with injective coding)
OUTPUT:
dict, key > key
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition({4:p, 1:q, 2:r}) sage: Q = PolyhedronPartition({0:p, 5:q}) sage: d = P.keys_permutation(Q) sage: d {1: 5, 2: 1, 4: 0} sage: P.rename_keys(d) Polyhedron partition of 3 atoms with 3 letters

keys_permutation_lexicographic
()¶ Return a permutation relabelling of the keys for self in increasing order for the lexicographic order of the centers of the polyhedrons.
OUTPUT:
dict, key > key
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition({4:p, 1:q, 2:r}) sage: d = P.keys_permutation_lexicographic() sage: d {1: 1, 2: 2, 4: 0} sage: P.rename_keys(d) Polyhedron partition of 3 atoms with 3 letters
sage: Q = PolyhedronPartition({0:p, 5:q}) sage: Q.keys_permutation_lexicographic() {0: 0, 5: 1}
It works when the partition has two atoms coded by the same key:
sage: P = PolyhedronPartition([(0,p), (0,q), (3,r)]) sage: d = P.keys_permutation_lexicographic() sage: d {0: 0, 3: 1} sage: P.rename_keys(d).alphabet() {0, 1}

merge_atoms
(d)¶ Return the polyhedron partition obtained by merging atoms having the same image under the dictionnary and such that their union is convex.
INPUT:
d
– dict
OUTPUT:
a polyhedron partition
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition({0:p, 1:q, 2:r}) sage: P.merge_atoms({0:4, 1:4, 2:5}) Polyhedron partition of 2 atoms with 2 letters sage: P.merge_atoms({0:4, 1:5, 2:4}) Polyhedron partition of 3 atoms with 2 letters
When pair of atoms are not convex, it needs to merge 3 or more atoms:
sage: h = 1/5 sage: p = Polyhedron([(0,0),(h,1h),(0,1)]) sage: q = Polyhedron([(0,1), (h,1h), (1,1)]) sage: r = Polyhedron([(0,0), (h,1h), (1,1), (1,0)]) sage: P = PolyhedronPartition({0:p, 1:q, 2:r}) sage: P.merge_atoms({0:4, 1:4, 2:4}) Polyhedron partition of 1 atoms with 1 letters

plot
(label_pos='insphere_center')¶ INPUT:
label_pos
– string (default:'insphere_center'
) or'center'
for the center of the polytope
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: P.plot() Graphics object consisting of 21 graphics primitives

refine_by_hyperplane
(ieq)¶ Refine the partition with the two half spaces of each side of an hyperplane.
INPUT:
ieq
– list, an inequality. An entry equal to “[1,7,3,4]” represents the inequality 7x_1+3x_2+4x_3>= 1.
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/3 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (h,0)]) sage: r = Polyhedron([(h,1), (1,1), (1,h), (h,0)]) sage: s = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s}) sage: ieq = [4, 5, 1] sage: P.refine_by_hyperplane(ieq) Polyhedron partition of 6 atoms with 6 letters

refinement
(other, certificate=False)¶ Return the polyhedron partition obtained by the intersection of the atoms of self with the atoms of other.
Only atoms of positive volume are kept.
INPUT:
other
– a polyhedron partitioncertificate
– boolean (default:False
), return a dictionnary for i:(p,q) if atom number i was obtained as the intersection of atoms p in self and q in other
OUTPUT:
a polyhedron partition
or
(polyhedron partition, dict)
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/3 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (h,0)]) sage: r = Polyhedron([(h,1), (1,1), (1,h), (h,0)]) sage: s = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s}) sage: g = 1/5 sage: t1 = Polyhedron([(g,g), (g,1g), (1g,g) ]) sage: t2 = Polyhedron([(g,1g), (1g,g), (1g,1g)]) sage: Q = PolyhedronPartition([t1,t2]) sage: P.refinement(Q) Polyhedron partition of 8 atoms with 8 letters

rename_keys
(d)¶ Return a polyhedron partition whose keys are the images under a map.
INPUT:
d
– dict, function old key > new key
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: Q = P.rename_keys({0:'b', 1:'a', 2:'z'}) sage: Q Polyhedron partition of 3 atoms with 3 letters sage: sorted(key for key,p in Q) ['a', 'b', 'z']
It does not have to be injective:
sage: Q = P.rename_keys({0:'b', 1:'a', 2:'b'}) sage: sorted(key for key,p in Q) ['a', 'b', 'b']

tikz
(fontsize='\\\\normalsize', scale=1, label_format='{}', label_pos='insphere_center', extra_code='')¶ INPUT:
fontsize
– string (default:r'\normalsize'
scale
– number (default:1
)label_format
– string (default:r'{}'
) to be called withlabel_format.format(key)
label_pos
– string (default:'insphere_center'
) or'center'
for the center of the polytopeextra_code
– string (default:''
)
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: _ = P.tikz().pdf(view=False)
Irrational partition:
sage: z = polygen(QQ, 'z') #z = QQ['z'].0 # same as sage: K = NumberField(z**2z1, 'phi', embedding=RR(1.6)) sage: phi = K.gen() sage: h = 1/phi^2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (h,0)]) sage: r = Polyhedron([(h,1), (1,1), (1,h), (h,0)]) sage: s = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s}, base_ring=K) sage: _ = P.tikz().pdf(view=False)
Testing the options:
sage: _ = P.tikz(fontsize=r'\scriptsize').pdf(view=False) sage: _ = P.tikz(scale=2).pdf(view=False) sage: _ = P.tikz(label_format=r'$a_{{{}}}$').pdf(view=False)

translate
(displacement)¶ Return the translated partition of polyhedron.
INPUT:
displacement
– a displacement vector or a list/tuple of coordinates that determines a displacement vector.
OUTPUT:
The translated partition.
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: P.translate((1,1)) Polyhedron partition of 3 atoms with 3 letters

volume
()¶ EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: P.volume() 1
TESTS:
sage: PolyhedronPartition([], base_ring=ZZ).volume() 0

volume_dict
(normalize=False)¶ INPUT
normalize
– boolean (default:False
), whether to normalize the sum of the whole volume to 1
EXAMPLES:
sage: from slabbe import PolyhedronPartition sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: P = PolyhedronPartition([p,q,r]) sage: P.volume_dict() {0: 1/8, 1: 3/4, 2: 1/8} sage: (2*P).volume_dict() {0: 1/2, 1: 3, 2: 1/2}

slabbe.polyhedron_partition.
center_insphere_polytope
(polytope, solver=None)¶ Return the center and radius of maximal inscribed sphere
INPUT:
polytope
– polytope
OUTPUT:
a 2tuple (center, radius)
EXAMPLES:
sage: from slabbe.polyhedron_partition import center_insphere_polytope sage: P = polytopes.associahedron(['A',3]) sage: center_insphere_polytope(P) # abs tol 1e6 ([0.03553390593273766, 0.5355339059327378, 0.03553390593273766], 1.4644660940672622) sage: center_insphere_polytope(P + vector((10,10,10))) # abs tol 1e6 ([10.035533905932738, 10.535533905932738, 10.035533905932738], 1.4644660940672622)
sage: P = Polyhedron([(0,0), (1,0), (0,10)]) sage: center_insphere_polytope(P) # abs tol 1e6 ([0.47506218943955486, 0.47506218943955486], 0.47506218943955486)

slabbe.polyhedron_partition.
find_unused_key
(d, sequence)¶ Return the first key in sequence which is not in d.
EXAMPLES:
sage: from slabbe.polyhedron_partition import find_unused_key sage: d = {3:32, 0:21, 1:4, 5:5} sage: find_unused_key(d, NN) 2 sage: d[2] = 1234 sage: find_unused_key(d, NN) 4 sage: d[4] = 1234 sage: find_unused_key(d, NN) 6

slabbe.polyhedron_partition.
is_union_convex
(t)¶ Return whether the union of the polyhedrons is convex.
INPUT:
t
– list of polyhedron
EXAMPLES:
sage: from slabbe.polyhedron_partition import is_union_convex sage: h = 1/2 sage: p = Polyhedron([(0,h),(0,1),(h,1)]) sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)]) sage: r = Polyhedron([(h,0), (1,0), (1,h)]) sage: is_union_convex((p,q,r)) True sage: is_union_convex((p,q)) True sage: is_union_convex((p,r)) False
Here we need to consider the three at the same time to get a convex union:
sage: h = 1/5 sage: p = Polyhedron([(0,0),(h,1h),(0,1)]) sage: q = Polyhedron([(0,1), (h,1h), (1,1)]) sage: r = Polyhedron([(0,0), (h,1h), (1,1), (1,0)]) sage: is_union_convex((p,q)) False sage: is_union_convex((p,r)) False sage: is_union_convex((q,r)) False sage: is_union_convex((p,q,r)) True