A convex cone is *polyhedral* if it is a finite intersection of halfspaces. A convex cone is * finitely generated * if it is the set of all nonnegative linear combinations of a finite set of vectors. The fundamental theorem for cones states that a convex cone is polyhedral if and only if it is finitely generated.

`FourierMotzkin` is a Macaulay2 implementation of the Double Description Method (of Fourier, Dines and Motzkin) for converting between these two basic representations for convex cones. For polytopes, this allows one to convert between the convex hull of a finite point set and the bounded intersection of halfspaces.

Here are some examples illustrating some uses of this package.

- Finding the facets of the cyclic polytope
- Finding the facets of the permutahedron
- Applications to multigraded polynomial rings

This package is intended for use with relatively small polyhedra. For larger polyhedra, please consider cdd, lrs or qhull; also see polymake.

For an introduction to polyhedra and Fourier-Motzkin elimination, we recommend Chapter 2 in Gunter M. Ziegler's *Lectures on Polytopes*, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995. For historical comments, see Section 12.2 in Alexander Schrijver's *Theory of Linear and Integer Programming* Wiley-Interscience Series in Discrete Mathematics, John Wiley and Sons, Chichester, 1986.

We thank Rene Birkner for help debugging the package.

- Functions and commands
- fourierMotzkin -- interchange inequality/generator representation of a polyhedral cone