Contents

Introduction

Convex is a Maple package for computations in rational convex geometry. Here "rational" means that all coordinates must be rational numbers. The package provides functions for "linear" as well as "affine" convex geometry.

In the affine setting, the basic objects are polyhedra, which are intersections of finitely many (affine) halfspaces. Polyhedra can also be described as the convex hull of finitely many points and rays. A bounded polyhedron is also called a polytope. In the Convex package, polyhedra are represented by the type POLYHEDRON. They may contain lines and may not be full-dimensional. The most important functions to define a POLYHEDRON are POLYHEDRON[convhull] and POLYHEDRON[intersection].

The linear setting is based on cones, which are intersections of finitely many linear halfspaces (i.e., whose boundary contains the origin). They are generated by finitely many rays. In the Convex package, cones are represented by the type CONE. They may contain lines and may not be full-dimensional. A CONE can be created from either description with the functions CONE[poshull] and CONE[intersection], respectively.

The Convex package can deal with polyhedral complexes (simplicial complexes, for example), and fans. See PCOMPLEX and FAN. It also provides functions to do calculations in the face lattice of a cone or polyhedron, see the types CFACE and PFACE. The functions traverse and traverse2 are some kind of map for faces: One can apply a given function to all faces of a cone or polyhedron, or to all pairs (f1, f2), where f1 is a facet of f2. See CONE[traverse], CONE[traverse2] and POLYHEDRON[traverse], POLYHEDRON[traverse2]. More generally, these functions can be used with fans and polyhedral complexes.

Author

Convex was written by Matthias Franz, based on work of others as indicated. Copyright © 1999-2004 Matthias Franz. It is distributed under the GNU General Public License.

Credits

The package uses an implementation of the double description method adapted from Doran Wilde's polyhedral library.

The author thanks Annette A'Campo-Neuen, Gottfried Barthel, Frédéric Béringer, Marion Dieudonné, Jürgen Hausen, Michael Nüsken, Bruno Salvy, Thorsten Theobald and Marcel Widmann for helpful suggestions and bug reports.

Citations

If you publish a mathematical result that has been partly obtained using the Convex package, please cite the package, just as you would cite any other paper. Specifically, please refer to: Matthias Franz: Convex - a Maple package for convex geometry, version 1.0 (2004), available via internet at http://www-fourier.ujf-grenoble.fr/~franz/convex/. I would also appreciate if you could inform me about such a paper.