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Quantum Jumps of Normal Polytopes

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2016

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Bruns, Winfried
Gubeladze, Joseph

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https://doi.org/10.1007/s00454-016-9773-7
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Discrete & Computational Geometry. Springer. 2016, 56(1), pp. 181-215. ISSN 0179-5376. eISSN 1432-0444. Available under: doi: 10.1007/s00454-016-9773-7

Zusammenfassung

We introduce a partial order on the set of all normal polytopes in Rd. This poset NPol(d) is a natural discrete counterpart of the continuum of convex compact sets in Rd, ordered by inclusion, and exhibits a remarkably rich combinatorial structure. We derive various arithmetic bounds on elementary relations in NPol(d), called quantum jumps. The existence of extremal objects in NPol(d) is a challenge of number theoretical flavor, leading to interesting classes of normal polytopes: minimal, maximal, spherical. Minimal elements in 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 NPol(4) and NPol(5), found by a combination of the developed theory, random generation, and extensive computer search.

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Fachgebiet (DDC)
510 Mathematik

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Lattice polytope, Normal polytope, Maximal polytope, Quantum jump

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ISO 690BRUNS, Winfried, Joseph GUBELADZE, Mateusz MICHALEK, 2016. Quantum Jumps of Normal Polytopes. In: Discrete & Computational Geometry. Springer. 2016, 56(1), pp. 181-215. ISSN 0179-5376. eISSN 1432-0444. Available under: doi: 10.1007/s00454-016-9773-7
BibTex
@article{Bruns2016Quant-52322,
  year={2016},
  doi={10.1007/s00454-016-9773-7},
  title={Quantum Jumps of Normal Polytopes},
  number={1},
  volume={56},
  issn={0179-5376},
  journal={Discrete & Computational Geometry},
  pages={181--215},
  author={Bruns, Winfried and Gubeladze, Joseph and Michalek, Mateusz}
}
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