Integer decomposition of smooth polytopes

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Is it true that every smooth polytope has the integer decomposition property?


This question was asked by Tadao Oda [1], see also the survey of Gubeladze [2]. The conjecture has been verified for smooth polytopes containing at most 12 lattice points [3], and there is no known smooth polytope without a unimodular triangulation. For cases when a unimodular triangulation is known to exist, see [4].


  1. T. Oda, Problems on Minkowski sums of convex lattice polytopes, arXiv link
  2. J. Gubeladze, Normal polytopes, manuscript
  3. T. Bogart et al., Few smooth d-polytopes with n lattice points, arXiv link
  4. C. Haase, A. Paffenholz, L.C. Piechnik, F. Santos, Existence of unimodular triangulations - positive results, arXiv link