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Polyhedra -- for computations with convex polyhedra, cones, and fans

Description

A rational convex Polyhedron is the intersection of finitely many affine half-spaces over QQ or equivalently, the convex hull of a finite set of vertices and rays. A rational convex polyhedral Cone is the intersection of finitely many linear half-spaces over QQ or equivalently, the positive hull of a finite set of rays. A Fan is a finite collection of cones such that for each cone all its faces are in the fan and for two cones in the fan the intersection is a face of each.

Polyhedra uses the FourierMotzkin package by Gregory G. Smith. Each polyhedron or cone is saved in both descriptions and a fan is saved as the list of its generating cones.

Here are some examples illustrating the main uses of this package.

For an introduction to polyhedra and cones, we recommend Gunter M. Ziegler's Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995.

The author would like to thank Nathan Ilten for contributing several functions to the package.

Author

Certification

Version 1.0.5 of this package was accepted for publication in volume 1 of the journal The Journal of Software for Algebra and Geometry: Macaulay2 on 2009-09-07, in the article Polyhedra: a package for computations with convex polyhedral objects. That version can be obtained from the journal or from the Macaulay2 source code repository, after installing subversion, with the following shell command:

   svn export -r 9344 svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/Polyhedra.m2

The following command will display the log messages accompanying any changes to the file in the repository since publication.

   svn log -r 9345:HEAD svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/Polyhedra.m2

The following command will summarize the changes to the file in the repository since publication, in the format the program diff uses: lines starting with + have been added, and lines starting with - have been removed. (Changes to white space or end of line style will not be reported.)

   svn diff -x "-b --ignore-eol-style" -r 9344:HEAD svn://macaulay2.math.uiuc.edu/Macaulay2/trunk/M2/Macaulay2/packages/Polyhedra.m2

The differences between two releases in the repository mentioned in the log can be displayed by replacing 9344:HEAD by the pair of release numbers separated by a colon.

Version

This documentation describes version 1.0.8 of Polyhedra.

Source code

The source code from which this documentation is derived is in the file Polyhedra.m2.

Exports