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.
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, svn://svn.macaulay2.com/Macaulay2/trunk/M2/Macaulay2/packages/Polyhedra.m2, release number 9344.
This documentation describes version 1.3 of OldPolyhedra.
The source code from which this documentation is derived is in the file OldPolyhedra.m2.