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=== Sage 8.0 === Here is a sample of changes related to Polyhedral Geometry in this release. * Highlights: * Integration with Latte (function .integrate() ) * New class for Voronoi Diagrams * Backend polymake for polyhedron * Improvements: * Fixed CartesianProduct with distinct dimensions * Fixed an error in .is_lattice_polytope() * Fixed overflow in integral_points() * Fixed normaliz backend with empty polyhedron * New methods for Polyhedron: * integral_points_count() === Sage 7.6 === Here is a sample of changes related to Polyhedral Geometry in this release: * Improvements: * documentation of Combinatorial and Discrete Geometry * combinatorial_automorphism_group * Fixes on .volume() * Fixes in integral_points() * New methods for Polyhedron: * .is_inscribed() * .neighborliness() * .is_neighborly() * .normal_fan() * .face_fan() * .is_combinatorially_isomorphic() * .face_truncation() * New methods for SimplicialComplexes: * star of a face * stellar subdivision * Generic interface to latte integrale: * count points 
A Roadmap for Optimization and Polyhedral Geometry softwares interactions
(around Sage, (PyQ)Normaliz, SCIP, eantic, Polymake, and others)
This page is intended to describe the current status of development of Sage related to optimization and polyhedral geometry.
Future & Current Projects
 Extend the usage of Normaliz in Sage
 Extend the polyhedral geometry component of sage
 Polyhedral arrangements and complexes
Meetings
Sage Days 84 on Polytopes in Olot
Tickets
Here are some practical lists of tickets:
Tickets from the Sage Days 84 March 2017
Tickets from the IMA Coding Sprint  April 2018
Current tickets of the Geometry component
Metatickets
There are already a few metatickets related to this topic:
Polytopes, lattice (integer) point counting / enumeration, and their applications
Improvements to MixedIntegerLinearProgram, its backends, and InteractiveLinearProgram
To do
Test the algebraic polytopes #25097
Make Polyhedron_normaliz accept algebraic stuff, related to #25097 (adapt _init_from_Hrepresentation)
 Add known irrational polytope to the library of polytopes
repr of NumberFields (the parents) should indicate its embedding if there is one #21161 (this ticket will help in debugging and have good feedback)
 Thematic Tutorial for Py(Q)Normaliz installation/Usage in Sage
 Increase documentation in Py(Q)Normaliz
Deprecate the opacity argument for plot3d and use alpha #21984
Make Ehrhart (quasi)polynomial accessible through normaliz (and consisten with the current one) #18957
Adjust the new features of the normaliz backend to the latest version #25091 #25090
Progress Report Timeline
April 2018
Sage
Here are improvements realized to objects related to polyhedron in Sage:
Thematic tutorial on polyhedron in Sage #22572
Make features of normaliz accessible #25091
number_field_elements_from_algebraics should create embedded number field elements #20181 Sagedevel discussion
Helper functions and improvements in the normaliz backend #25097
Change error message in construction of polyhedron object #24835
Improve the output of repr_pretty for Polyhedron #24837
Implement stacking onto a face of a polyhedron #24847
Implement join of polytopes #24848
Implement one point suspension and face split of polyhedron #24886
The polar of a polyhedron should carry the backend used #25081
Construct RationalPolyhedralFan from possibly overlapping cones #25122
polygon3d ignores the "alpha" (and equivalent "opacity") argument #25095
Update normaliz 3.5.3 and pynormaliz 1.12 in Sage. Now Positive review. #22984
PolyhedralComplex: preliminary implementation, see #20877
(Q)Normaliz
http://normaliz.uniosnabrueck.de
Upgrade normaliz to 3.5.3 and pynormaliz to 1.12 #22984: repair of performance loss
Upgrade normaliz to 3.5.4 and pynormaliz to 1.16 #25090: Ehrhart series and triangulation of polytope also for inhomogeneous input
Algebraic polytopes in Qnormaliz using eantic #25097: convex hull computation completed, improvement of Qnormaliz input and output, control of interrupts and parallelization via libQnormaliz
fixed: getProjectionCone might segfault https://github.com/Normaliz/Normaliz/issues/180
fixed: BadInputException for ProjectCone https://github.com/Normaliz/Normaliz/issues/181
Unified getter https://github.com/Normaliz/Normaliz/pull/182available for interfaces (return of results by type instead of computation goal)
Py(Q)Normaliz
https://pypi.python.org/pypi/PyNormaliz https://github.com/Normaliz/PyNormaliz
Upgrade normaliz to 3.5.3 and pynormaliz to 1.12 #22984
Upgraded pynormaliz to 1.16 https://github.com/Normaliz/PyNormaliz
Started PyQNormaliz. https://github.com/sebasguts/PyQNormaliz
PyScipOpt
https://github.com/SCIPInterfaces/PySCIPOpt
Sage devel discussion
Polymake
Upgrade polymake to version 3.2r2 (current interface breaks) #24905
Sage 8.0
Here is a sample of changes related to Polyhedral Geometry in this release.
 Highlights:
 Integration with Latte (function .integrate() )
 New class for Voronoi Diagrams
 Backend polymake for polyhedron
 Improvements:
Fixed CartesianProduct with distinct dimensions
 Fixed an error in .is_lattice_polytope()
 Fixed overflow in integral_points()
 Fixed normaliz backend with empty polyhedron
 New methods for Polyhedron:
 integral_points_count()
Sage 7.6
Here is a sample of changes related to Polyhedral Geometry in this release:
 Improvements:
 documentation of Combinatorial and Discrete Geometry
 combinatorial_automorphism_group
 Fixes on .volume()
 Fixes in integral_points()
 New methods for Polyhedron:
 .is_inscribed()
 .neighborliness()
 .is_neighborly()
 .normal_fan()
 .face_fan()
 .is_combinatorially_isomorphic()
 .face_truncation()
New methods for SimplicialComplexes:
 star of a face
 stellar subdivision
 Generic interface to latte integrale:
 count points
A list of other Polyhedral Geometry softwares
What are other softwares dealing with polyhedral geometry? Here is a nonexhaustive list.
 Maple:
https://www.maplesoft.com/support/help/Maple/view.aspx?path=PolyhedralSets%2fPolyhedralSet (seems to be only rational and implemented in 2015).
http://qhull.org/ (seems old?)
https://www.maplesoft.com/products/maple/new_features/Maple2018/ComputationalGeometry.aspx
https://www.maplesoft.com/support/help/Maple/view.aspx?path=PolyhedralSets
 CGAL:
 A paper:
Jing & Maza https://dl.acm.org/citation.cfm?id=3177798
 Matlab:
 Matematica: