Beneath-and-Beyond Revisited

  • Michael Joswig
Conference paper


It is shown how the Beneath-and-Beyond algorithm can be used to yield another proof of the equivalence of V- and H-representations of convex polytopes. In this sense this paper serves as the sketch of an introduction to polytope theory with a focus on algorithmic aspects. Moreover, computational results are presented to compare Beneath-and-Beyond to other convex hull implementations.


Convex Hull Simplicial Complex Simplicial Polytopes Reverse Search Memory Overflow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    D. Avis. A revised implementation of the reverse search vertex enumeration algorithm. In Polytopes—combinatorics and computation (Oberwolfach, 1997), volume 29 of DMV Sem., pages 177–198. Birkhäuser, Basel, 2000.Google Scholar
  2. 2.
    D. Avis. lrs, Irslib, Version 4.1., 2001.
  3. 3.
    D. Avis, D. Bremner, and R. Seidel. How good are convex hull algorithms? Comput. Geom., 7(5–6): 265 301, 1997.Google Scholar
  4. 4.
    D. Avis and K. Fukuda. A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Comput. Geom.,8(3):295313, 1992.Google Scholar
  5. 5.
    L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and real computation. Springer-Verlag, New York, 1998. With a foreword by Richard M. Karp.Google Scholar
  6. 6.
    D. Bremner. Incremental convex hull algorithms are not output sensitive. Discrete Comput. Geom., 21 (1): 57–68, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    A. Brondsted. An introduction to convex polytopes, volume 90 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1983.Google Scholar
  8. 8.
    CGAL, Version 2.4.,2002.
  9. 9.
    J.-G. Dumas, F. Heckenbach, D. Saunders, and V. Welker. Computing simplicial homology based on efficient smith normal form algorithms. In this volume,pages 177–206.Google Scholar
  10. 10.
    H. Edelsbrunner. Algorithms in combinatorial geometry. Springer-Verlag, Berlin, 1987.zbMATHCrossRefGoogle Scholar
  11. 11.
    K. Fukuda. cdd±, Version 0.76a., 2001.
  12. 12.
    K. Fukuda. cddlib, Version 0.92a., 2001.
  13. 13.
    K. Fukuda and A. Prodon. Double description method revisited. In Combinatorics and computer science (Brest, 1995), volume 1120 of Lecture Notes in Comput. Sci., pages 91–111. Springer, Berlin, 1996.Google Scholar
  14. 14.
    E. Gawrilow and M. Joswig. polymake, version 1.5.1: a software package for analyzing convex polytopes., 1997–2003.
  15. 15.
    E. Gawrilow and M. Joswig. polymake: a framework for analyzing convex polytopes. In G. Kalai and G. M. Ziegler, editors, Polytopes - Combinatorics and Computation, pages 43–74. Birkhäuser, 2000.Google Scholar
  16. 16.
    E. Gawrilow and M. Joswig. polymake: an approach to modular software design in computational geometry. In Proceedings of the 17th Annual Symposium on Computational Geometry,pages 222–231. ACM, 2001. June 3–5, 2001, Medford, MA.Google Scholar
  17. 17.
    J.E. Goodman, R. Pollack, and B. Sturmfels. The intrinsic spread of a configuration in P d . J. Amer. Math. Soc. 3 (3): 639–651, 1990.MathSciNetzbMATHGoogle Scholar
  18. 18.
    GNU multiprecision library, Version 4.1. http://www.
  19. 19.
    M. Grötschel. Optimierungsmethoden I. Technical report, Universität Augsburg, 1985. Skriptum zur Vorlesung im WS 1984/85.Google Scholar
  20. 20.
    B. Grünbaum. Convex polytopes. Springer, 2003. 2nd edition edited by V. Kaibel, V. Klee, and G.M. Ziegler.Google Scholar
  21. 21.
    M. Haiman. A simple and relatively efficient triangulation of the n-cube. Discrete Comput. Geom., 6 (4): 287–289, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    M. Joswig and G. M. Ziegler. Convex hulls, oracles, and homology. Preprint, 11 pages, arXiv: math.MG/0301100.Google Scholar
  23. 23.
    V. Kaibel and M. E. Pfetsch. Some algorithmic problems in polytope theory. In this volume,pages 23–47.Google Scholar
  24. 24.
    LEDA, Version 4.3. Algorithmic Solution Software GmbH,
  25. 25.
    C. W. Lee. Subdivisions and triangulations of polytopes. In J. Goodman and J. O’Rourke, editors, Handbook of Discrete and Computational Geometry, pages 271–290. CRC Press, 1997.Google Scholar
  26. 26.
    J. Matoušek. Lectures on Discrete Geometry. Springer, 2002.Google Scholar
  27. 27.
    J. Pfeifle and J. Rambau. Computing triangulations using oriented matroids. In this volume, pages 49–75.Google Scholar
  28. 28.
    K. Polthier, S. Khadem, E. Preuss, and U. Reitebuch. JavaView, Version 2.21.
  29. 29.
    A. Schrijver. Theory of linear and integer programming. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons Ltd., Chichester, 1986.Google Scholar
  30. 30.
    C. Yap and Z. Du. Core Library (CORE), Version 1.4., 2002.
  31. 31.
    G. M. Ziegler. Lectures on Polytopes. Springer, 1998. 2nd ed.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Michael Joswig
    • 1
  1. 1.Institut für Mathematik, MA 6-2Technische Universität BerlinBerlinGermany

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