Annals of Combinatorics

, Volume 23, Issue 2, pp 255–262 | Cite as

Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP

  • Matthias Beck
  • Christian Haase
  • Akihiro Higashitani
  • Johannes HofscheierEmail author
  • Katharina Jochemko
  • Lukas Katthän
  • Mateusz Michałek


In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda’s conjecture for centrally symmetric 3-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices.


Smooth lattice polytopes Integer decomposition property Oda’s conjecture Central symmetry 3-dimensional polytopes 

Mathematics Subject Classification

52B20 52B10 52B12 



The authors would like to thank the Mathematisches Forschungsinstitut Oberwolfach for hosting the Mini-Workshop Lattice polytopes: methods, advances and applications in Fall 2017 during which this project evolved. We are grateful to Joseph Gubeladze, Bernd Sturmfels, and two anonymous referees for helpful comments. Katharina Jochemko was supported by the Knut and Alice Wallenberg foundation. Lukas Katthän was supported by the DFG, grant KA 4128/2-1. Mateusz Michałek was supported by the Polish National Science Centre grant no. 2015/19/D/ST1/01180. The work on this paper was completed while the fifth and sixth authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 Semester.


  1. 1.
    Bruns, W., Gubeladze, J.: Normality and covering properties of affine semigroups. J. Reine Angew. Math. 510, 161–178 (1999)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bruns, W., Gubeladze, J.: Polytopes, Rings, and \(K\)-Theory. Springer Monographs in Mathematics. Springer, Dordrecht (2009)Google Scholar
  3. 3.
    Castillo, F., Liu, F., Nill, B., Paffenholz, A.: Smooth polytopes with negative Ehrhart coefficients. J. Combin. Theory Ser. A 160, 316–331 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Huber, B., Rambau, J., Santos, F.: The Cayley trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings. J. Eur. Math. Soc. (JEMS) 2(2), 179–198 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Oda, T.: Problems on Minkowski sums of convex lattice polytopes. arXiv:0812.1418 (2008)
  6. 6.
    Rambau, J: Polyhedral subdivisions and projections of polytopes. Ph.D. Thesis. Technical University of Berlin (1996)Google Scholar
  7. 7.
    Schneider, R.: Convex Bodies: the Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications, Vol. 44. Cambridge University Press, Cambridge (1993)Google Scholar
  8. 8.
    Sturmfels, B.: On the Newton polytope of the resultant. J. Algebraic Combin. 3(2), 207–236 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Matthias Beck
    • 1
  • Christian Haase
    • 2
  • Akihiro Higashitani
    • 3
  • Johannes Hofscheier
    • 4
    Email author
  • Katharina Jochemko
    • 5
  • Lukas Katthän
    • 6
  • Mateusz Michałek
    • 7
    • 8
  1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
  2. 2.MathematikFreie Universität BerlinBerlinGermany
  3. 3.Department of MathematicsKyoto Sangyo UniversityKyotoJapan
  4. 4.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  5. 5.Department of MathematicsRoyal Institute of TechnologyStockholmSweden
  6. 6.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  7. 7.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  8. 8.Mathematical Institute of the Polish Academy of SciencesWarszawaPoland

Personalised recommendations