Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP
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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.
KeywordsSmooth lattice polytopes Integer decomposition property Oda’s conjecture Central symmetry 3-dimensional polytopes
Mathematics Subject Classification52B20 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.
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