Opened 4 years ago
Last modified 4 years ago
#27534 closed enhancement
Implement Lawrence extension for polytopes — at Version 13
Reported by:  Laith Rastanawi  Owned by:  

Priority:  major  Milestone:  sage8.8 
Component:  geometry  Keywords:  polytopes, lawrence extension 
Cc:  JeanPhilippe Labbé  Merged in:  
Authors:  Laith Rastanawi  Reviewers:  Jonathan Kliem 
Report Upstream:  N/A  Work issues:  
Branch:  public/27534 (Commits, GitHub, GitLab)  Commit:  69cd2c3f76444244edc5fa2d9d38001ac695ba36 
Dependencies:  Stopgaps: 
Description (last modified by )
Adding the following methods to base.py:
lawrence_extension(self, v)
: Return the Lawrence extension ofself
on the pointv
.lawrence_polytope(self)
: Return the Lawrence polytope ofself
.is_lawrence_polytope(self)
: Returntrue
ifself
is a Lawrence polytope.
for definitions, see Section 6.6 of Lectures on Polytopes, Günter M. Ziegler, [Zie2007]
Change History (13)
comment:1 Changed 4 years ago by
Branch:  → public/27534 

Commit:  → a7849b3ecea4f36f7233940ee6d18a0af943013c 
comment:2 Changed 4 years ago by
Status:  new → needs_review 

comment:3 Changed 4 years ago by
Reviewers:  → Jonathan Kliem 

Status:  needs_review → needs_work 
A few small comments.
 Single quotes (or whatever the name of those things is) is for latex, double for codestyle. There is at least one place (using
self
), where this is inconsistent. The Lawrence extension of P
, what is P? How aboutThe Lawrence extension of a polytope P
? A comment I just received myself: "These bullet points for
INPUT:
do not, by general Sage convention, end in a period/fullstop."  It seems to me that the requirement of being full dimensional is not necessary. Instead one can require 0 to be contained, if its not full dimensional. If a polytope
P
satisfies "ax = 0" for all x in P, then the lawrence extension will just satisfie "(a,0)x = 0". I.e. if all hyperplanes containing P are linear, then the Lawrence extension will be contained in the "same" hyperplanes. I propose something like: if not self.is_full_dimensional():  raise NotImplementedError("`self` must be full dimensional") + if not self.is_full_dimensional(): + if not self.contains([0]*self.ambient_dim()): + raise NotImplementedError("``self`` must be fulldimensional or contain the origin, try ``self.translation``")
Then, one can add an example that explains how to proceed in case of a not fulldimensional polytope:
sage: P = polytopes.permutahedron(4) sage: Q = lawrence_polytope(P) Traceback (most recent call last): ... NotImplementedError: ``self`` must be fulldimensional or contain the origin, try ``self.translation`` sage: T = P.translation(vector(P.vertices()[0])) sage: Q = lawrence_polytope(T) sage: Q A 26dimensional polyhedron in ZZ^28 defined as the convex hull of 47 vertices
Maybe add another example like this to the Lawrence extension (maybe Birkhoff_polytope(3)
).
 I find too many commands in one line hard to read:
 sage: P = polytopes.cube(); Q = P.lawrence_polytope(); Q.is_lawrence_polytope()  True + sage: P = polytopes.cube() + sage: Q = P.lawrence_polytope() + sage: Q.is_lawrence_polytope() + True
 I would use
True
instead oftrue
.  You never use
d = self.dim()
.
comment:4 Changed 4 years ago by
Thanks for your comments.
The functions need to be rewritten entirely: for instance, my function lawrence_polytope
does produce a Lawrence polytope, but not the Lawrence polytope of self
according to the usual definitions (there are two different definitions for it).
I will come back to this ticket later and fix everything.
comment:5 followups: 6 11 Changed 4 years ago by
One more remark.
is_lawrence_polytope
is very slow in cases, where the polytope does not have few vertices. Try checking this for permutahedron(6)
.
Instead of computing the Galetransform, one could just grap the information from the facets.
 Create a list of vertices
V
.  Iterate through all facets
F
containing exactlyn1
vertices.  Remove the vertex not contained in
F
fromV
(this vertex "is" the origin in the Gale transform).  Iterate through all facets
F
containing exactlyn2
vertices.  Let v_1,v_2 not be in
F
. If v_1 and v_2 inV
, remove them.
If at the end of this V
is empty, the Gale diagram is centrally symmetric. If not, then it is not.
comment:6 followups: 8 12 Changed 4 years ago by
Replying to ghkliem:
One more remark.
is_lawrence_polytope
is very slow in cases, where the polytope does not have few vertices. Try checking this forpermutahedron(6)
.
It is hard to have an algorithm that gets faster as the input gets larger!
In spite of that, yes, it might not be optimized, but I would not go into intricate implementation of the function unless there is yet a known use case.
One suggestion:
Perles' example of a nonrational polytope uses lawrence extension to be constructed. It would be nice to add the example to the library of polytopes (either using lawrence extension or just hardcoding the vertices. Or even better, hardcoding the vertices and in the test of the function for the polytope, construct the polytope using lawrence extension and check that it is really combinatorially isomorphic.
comment:7 Changed 4 years ago by
The Title of the ticket should be more precise about what the ticket provides.
In order to make this ticket more accessible, it would be nice to have related keywords: polytopes and lawrence extension.
comment:8 Changed 4 years ago by
Replying to jipilab:
Replying to ghkliem:
One more remark.
is_lawrence_polytope
is very slow in cases, where the polytope does not have few vertices. Try checking this forpermutahedron(6)
.It is hard to have an algorithm that gets faster as the input gets larger!
Sure. But its also nice to not recalculate something that is already known.
In spite of that, yes, it might not be optimized, but I would not go into intricate implementation of the function unless there is yet a known use case.
Whether a polytope is a Lawrence polytope can simply be decided from the incidence matrix. It is simply the question, whether the vertices V can be labeled v_{1},...,v_{n},w_{1},...,w_{2m} such that V\{v_{i}} and V\{w_{2j},w_{2j+1}} is a facet.
Moreover, the current version has issues. We discovered that gale_transform()
is not normalized. At the moment the answer depends on the realization of the polytope. Also, the current code will return True
for the bipyramid over a triangle. To my understanding this is not a Lawrence polytope as the Gale diagram is not symmetric with respect to multiplicity.
comment:9 Changed 4 years ago by
Is a pyramid over a Lawrence polytope a Lawrence polytope? I think the definition should allow for an arbitrary number of points at the origin of the Gale diagram.
polymake
's construction of a Lawrence polytope of a polytope with a vertex at the origin will even have an odd number of points. This construction is according to Bernd Sturmfels' definition, I think.
Let V be the vertex matrix of a polytope. The Lawrence polytope will have the following vertices according to Günter Ziegler (Lectures on Polytopes):
\begin{pmatrix} V & V \\ I_n & 2*I_n \end{pmatrix},
According to Bernd Sturmfels (https://arxiv.org/pdf/math/0202104.pdf):
\begin{pmatrix} V & 0 \\ I_n & I_n \end{pmatrix},
Both construction yield centrally symmetric Gale diagrams, but the second construction might not have an odd number of points, as pointed out.
Only the first construction will "add" for each point x in the Gale diagram a point x.
Btw, the first construction will not have gale_transform
be centrally symmetric, before normalizing. E.g. when applying it for the cube, those are the coordinates of the Gale transform:
sage: P = polytopes.cube() sage: I_n = matrix.identity(8) sage: V = P.vertices_matrix() sage: lambda_V = block_matrix([[V,V],[I_n, 2*I_n]]) sage: Polyhedron(lambda_V.transpose()).gale_transform() [(2, 0, 0, 0), (1, 0, 0, 0), (0, 2, 0, 0), (0, 1, 0, 0), (0, 0, 2, 0), (0, 0, 1, 0), (2, 2, 2, 0), (1, 1, 1, 0), (0, 0, 0, 2), (0, 0, 0, 1), (2, 2, 0, 2), (1, 1, 0, 1), (2, 0, 2, 2), (1, 0, 1, 1), (4, 2, 2, 2), (2, 1, 1, 1)]
comment:10 Changed 4 years ago by
Commit:  a7849b3ecea4f36f7233940ee6d18a0af943013c → 69cd2c3f76444244edc5fa2d9d38001ac695ba36 

Branch pushed to git repo; I updated commit sha1. New commits:
69cd2c3  improve is_lawrence and fix lawrence_polytope

comment:11 Changed 4 years ago by
Status:  needs_work → needs_review 

Replying to ghkliem:
One more remark.
is_lawrence_polytope
is very slow in cases, where the polytope does not have few vertices. Try checking this forpermutahedron(6)
.Instead of computing the Galetransform, one could just grap the information from the facets.
 Create a list of vertices
V
. Iterate through all facets
F
containing exactlyn1
vertices. Remove the vertex not contained in
F
fromV
(this vertex "is" the origin in the Gale transform). Iterate through all facets
F
containing exactlyn2
vertices. Let v_1,v_2 not be in
F
. If v_1 and v_2 inV
, remove them.If at the end of this
V
is empty, the Gale diagram is centrally symmetric. If not, then it is not.
This is really helpful and fast. I implemented this in is_lawrence_polytope
.
Both construction yield centrally symmetric Gale diagrams, but the second construction might not have an odd number of points, as pointed out.
I decided to only put the construction given by Ziegler. It works in the case where the polytope is not fulldimensional.
comment:12 Changed 4 years ago by
Replying to jipilab:
One suggestion:
Perles' example of a nonrational polytope uses lawrence extension to be constructed. It would be nice to add the example to the library of polytopes (either using lawrence extension or just hardcoding the vertices. Or even better, hardcoding the vertices and in the test of the function for the polytope, construct the polytope using lawrence extension and check that it is really combinatorially isomorphic.
This would be a very nice example. I think this needs its own ticket "adding Perles's example to polytopes library". After that we can add it to the test of the function lawrence_extension
.
comment:13 Changed 4 years ago by
Description:  modified (diff) 

Keywords:  polytopes lawrence extension added 
Summary:  Lawrence Extension → Implement Lawrence extension for polytopes 
New commits:
adding Lawrence polytope