Discrete & Computational Geometry

, Volume 44, Issue 1, pp 149–166 | Cite as

Totally Splittable Polytopes



A split of a polytope is a (necessarily regular) subdivision with exactly two maximal cells. A polytope is totally splittable if each triangulation (without additional vertices) is a common refinement of splits. This paper establishes a complete classification of the totally splittable polytopes.


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  1. 1.
    Bandelt, H.-J., Dress, A.: A canonical decomposition theory for metrics on a finite set. Adv. Math. 92(1), 47–105 (1992). MR 1153934 (93h:54022) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bayer, M.M.: Equidecomposable and weakly neighborly polytopes. Israel J. Math. 81(3), 301–320 (1993). MR 1231196 (94m:52015) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Billera, L.J., Gel’fand, I.M., Sturmfels, B.: Duality and minors of secondary polyhedra. J. Combin. Theory, Ser. B 57(2), 258–268 (1993). MR 1207491 (93m:52014) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 46. Cambridge University Press, Cambridge (1999). MR 1744046 (2000j:52016) MATHGoogle Scholar
  5. 5.
    De Loera, J., Rambau, J., Santos, F.: Triangulations: Structures and Algorithms. Springer, to appear Google Scholar
  6. 6.
    Gel’fand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhäuser, Boston (1994). MR 1264417 (95e:14045) CrossRefGoogle Scholar
  7. 7.
    Grünbaum, B.: Convex Polytopes, 2nd edn. Graduate Texts in Mathematics, vol. 221. Springer, New York (2003). Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. MR 1976856 (2004b:52001) Google Scholar
  8. 8.
    Herrmann, S., Jensen, A., Joswig, M., Sturmfels, B.: How to draw tropical planes. Electron. J. Combin. 16(2), R6 (2009–2010) MathSciNetGoogle Scholar
  9. 9.
    Herrmann, S., Joswig, M.: Splitting polytopes. Münster J. Math. 1, 109–141 (2008) MATHMathSciNetGoogle Scholar
  10. 10.
    Hirai, H.: A geometric study of the split decomposition. Discrete Comput. Geom. 36(2), 331–361 (2006). MR 2252108 (2007f:52025) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Shemer, I.: Neighborly polytopes. Israel J. Math. 43(4), 291–314 (1982). MR 693351 (84k:52008) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995). MR 1311028 (96a:52011) MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Fachbereich MathematikTU DarmstadtDarmstadtGermany
  2. 2.Institut für MathematikTU BerlinBerlinGermany

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