Convex - a Maple package for convex geometry

Current version: 1.1.3 (2009-03-07)

Copyright © 1999-2009 Matthias Franz

Convex is a Maple package for convex geometry. It can deal with polytopes and, more generally, with all kinds of polyhedra of (in principle) arbitrary dimension. The only restriction is that all coordinates must be rational.

The integration into the computer algebra system Maple makes Convex particularly suited for "applied" problems where polyhedra arise together with other mathematical structures. Examples we had in mind while writing the package were toric varieties (which are defined by fans) and moment polytopes related to representation theory. But of course the package is not restricted to these kinds of applications. On the other hand, if you want to break the current record for dualising polytopes, you will probably choose a different program.

The main design principles of Convex are as follows:

Ease of use

One quickly learns how to use it. (See the example.) Part of this strategy is some kind of "object-oriented approach": functions accept different types as input and automatically choose the right subroutine. Moreover, Convex is very easy to install (if you have already installed Maple).

Full generality

No restrictions are imposed on the polyhedra (apart from being rational). They may be unbounded or even contain lines. They may also be empty or not full-dimensional or live in 0-dimensional space. This is important in applications where one wants to apply functions to (many) polyhedra that are the result of previous calculations.

Separate treatment of polyhedra and cones

Although cones are special kinds of polyhedra, they are implemented as a separate type. One reason is that internally polyhedra are reduced to cones (by homogenisation). Another reason is that some results depend on whether a cone is considered as cone or as polyhedron. For example, the minimal face of a polyhedron is always the empty face, while the minimal face of a cone is the largest linear subspace it contains (its so-called "lineality space").

Other functions for polyhedra include: image and preimage under affine maps, Cartesian products and joins, tests for containment and other elementary properties, computations in the face lattice, drawing polytopes (up to dimension 3). One can apply a given function to all faces of a polyhedron, or to all pairs of faces (f1, f2) where f1 is a facet of f2. (Cf. the Maple function map, which applies a given function to all elements of a list or set.)

For cones, there are "linear" analogues of many functions for polyhedra, as well as others.

Convex can also deal with polyhedral complexes (in particular, with simplicial complexes) and with fans. Here functions include: test whether a fan is polytopal, again some map-like function to transverse a fan or polyhedral complex, integral and rational homology computation of polyhedral complexes.

Documentation

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Applications

Maple packages based on Convex

TorDiv by Florian Berchtold, Jürgen Hausen and Marcel Widmann

Divisors on toric varieties

Torhom by Matthias Franz

Homology of real and complex toric varieties

Interfaces

Research

• Matthias Franz, Moment polytopes of projective G-varieties and tensor products of symmetric group representations, J. Lie Theory 12 (2002), 539-549
• Annette A'Campo-Neuen and Jürgen Hausen, Orbit spaces of small tori, Resultate Math. 43 (2003), 13-22
• Frédéric Beringer and Françoise Richard-Jung, Multi-variate polynomials and Newton-Puiseux expansions, Springer LNCS 2630 (2003), 240-254
• Vladimir L. Popov, The cone of Hilbert nullforms, Proc. Steklov Inst. Math. 241 (2003), 177-194
• Kanchan Chandra and Cilanne Boulet, Ethnic diversity and democratic stability, preprint
• Etienne Rassart, Geometric approaches to computing Kostka numbers and Littlewood-Richardson coefficients, PhD thesis, MIT 2004
• Sara Billey, Victor Guillemin and Etienne Rassart, A vector partition function for the multiplicities of sl_kC, J. Algebra 278 (2004), 251-293
• Etienne Rassart, A polynomiality property for Littlewood-Richardson coefficients, J. Comb. Theory Ser. A 107 (2004), 161-179
• Giovanna Roda, Algorithms for change of orderings in the theory of Gröbner bases, PhD thesis, Universität Linz 2004
• Albrecht Klemm et al., Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections, JHEP 05 (2005) 023
• Guido Stehr, On the performance space exploration of analog integrated circuits, PhD thesis, TU München 2005
• Panagiotis Kaklis and Spyridon Dellas, On the shape effect of a control point: experimenting with NURBS surfaces, pp. 183-192 in: Tor Dokken and Bert Jüttler, Computational methods for algebraic spline surfaces, Springer, Berlin 2005
• John Edward Perry, Combinatorial criteria for Gröbner bases, PhD thesis, North Carolina State University (2005)
• Matthias Franz, The integral cohomology of toric manifolds, Proc. Steklov Inst. Math. 252 (2006), 53-62
• Florian Berchtold and Jürgen Hausen, GIT equivalence beyond the ample cone, Michigan Math. J. 54 (2006), 483-516
• Frédéric Bihan et al., Is every toric variety an M-variety?, Manuscripta Math. 120 (2006), 217-232
• G. Stehr, H. E. Graeb and K. J. Antreich, Analog performance space exploration by normal-boundary intersection and by Fourier-Motzkin elimination, IEEE Trans. CAD 26 (2007), 1733-1748
• Michael Kapovich, Shrawan Kumar and John J. Millson, The eigencone and saturation for Spin(8), Pure Appl. Math. Q. 5 (2009), 1-26
• Mark C. Wilson and Geoffrey Pritchard, Probability calculations under the IAC hypothesis, Math. Social Sci. 54 (2007), 244-256
• Valerie Hower, A counterexample to the maximality of toric varieties, Proc. Amer. Math. Soc. 136 (2008), 4139-4142
• Valerie Hower, Hodge spaces of real toric varieties, Collect. Math. 59 (2008), 215-237
• Robin Bergenthum et al., Process mining based on regions of languages, pp. 375-383 in: Gustavo Alonso, Peter Dadam and Michael Rosemann (eds.), Business process management, Springer, Berlin 2007
• Tomáš Kroupa, Geometry of possibility measures on finite sets, Int. J. Approx. Reasoning 48 (2008), 237-245
• Alexander A. Klyachko et al., Simple test for hidden variables in spin-1 systems, Phys. Rev. Lett. 101 (2008), 020403
• Magnus Dehli Vigeland, Smooth tropical surfaces with infinitely many tropical lines, preprint arXiv:math/0703682
• Carlos D'Andrea and Martín Sombra, The Newton polygon of a rational plane curve, preprint arXiv:0710.1103
• U. Derenthal, The nef cone volume in degrees 3 and 4, appendix to: R. de la Bretèche and T. D. Browning, Manin's conjecture for quartic del Pezzo surfaces with a conic fibration, preprint arXiv:0808.1616
• Jaron Treutlein, 3-dimensional lattice polytopes without interior lattice points, preprint arXiv:0809.1787
• Emmanuel Briand, Rosa Orellana and Mercedes Rosas, Quasipolynomial formulas for the Kronecker coefficients indexed by two two-row shapes, preprint arXiv:0812.0861
• Thomas Bliem, Weight multiplicities for so₅(C), pp. 80-86 in: Matthias Dehmer, Michael Drmota and Frank Emmert-Streib (eds.), Proceedings of the 2008 international conference on information theory and statistical learning, CSREA Press, 2008
• M. Studený and J. Vomlel, A geometric approach to learning BN structures, pp. 281-288 in: M. Jaeger, T. D. Nielsen (eds.), Proceedings of the Fourth European Workshop on Probabilistic Graphical Models (PGM'08), 2008
• Murat Altunbulak, The Pauli principle, representation theory, and geometry of flag varieties, PhD thesis, Bilkent University (2008)
• Murat Altunbulak and Alexander Klyachko, The Pauli principle revisited, Comm. Math. Phys. 282 (2008), 287-322
• A. I. Ginnis, E. I. Karousos, P. D. Kaklis, A discrete methodology for controlling the sign of curvature and torsion for NURBS, Computing 86 (2009), 117-129
• Ian Morrison and David Swinarski, Groebner techniques for low degree Hilbert stability, preprint arXiv:0910.2047
• Thomas Bliem, Chopped and sliced cones and representations of Kac-Moody algebras, J. Pure Appl. Algebra (2009), in press

Teaching

• Using computer algebra systems in mathematics (Michael Nüsken)
• IMMERSE 2007 algebra course (University of Nebraska)