Discrete & Computational Geometry

, Volume 56, Issue 1, pp 181–215 | Cite as

Quantum Jumps of Normal Polytopes

  • Winfried Bruns
  • Joseph Gubeladze
  • Mateusz Michałek


We introduce a partial order on the set of all normal polytopes in \(\mathbb R^d\). This poset \({{\mathsf {NPol}}}(d)\) is a natural discrete counterpart of the continuum of convex compact sets in \(\mathbb R^d\), ordered by inclusion, and exhibits a remarkably rich combinatorial structure. We derive various arithmetic bounds on elementary relations in \({{\mathsf {NPol}}}(d)\), called quantum jumps. The existence of extremal objects in \({{\mathsf {NPol}}}(d)\) is a challenge of number theoretical flavor, leading to interesting classes of normal polytopes: minimal, maximal, spherical. Minimal elements in \({{\mathsf {NPol}}}(5)\) have played a critical role in disproving various covering conjectures for normal polytopes in the 1990s. Here we report on the first examples of maximal elements in \({{\mathsf {NPol}}}(4)\) and \({{\mathsf {NPol}}}(5)\), found by a combination of the developed theory, random generation, and extensive computer search.


Lattice polytope Normal polytope Maximal polytope Quantum jump 

Mathematics Subject Classification

Primary 52B20 Secondary 11H06 52C07 



We thank B. van Fraassen for his comments in the early stages of this project. We are grateful to anonymous reviewers for their helpful comments and spotting several inaccuracies. Supported by Grants DFG BR 688/22-1 (Bruns), NSF DMS-1301487 and GNSF DI/16/5-103/12 (Gubeladze), Polish National Science Center Grant No. 2012/05/D/ST1/01063 (Michałek)


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institut für MathematikUniversität OsnabrückOsnabrückGermany
  2. 2.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
  3. 3.Institut für MathematikFreie UniversitätBerlinGermany
  4. 4.Polish Academy of SciencesWarsawPoland

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