Quantum Jumps of Normal Polytopes

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BRUNS, Winfried, Joseph GUBELADZE, Mateusz MICHALEK, 2016. Quantum Jumps of Normal Polytopes. In: Discrete & Computational Geometry. Springer. 56(1), pp. 181-215. ISSN 0179-5376. eISSN 1432-0444. Available under: doi: 10.1007/s00454-016-9773-7

@article{Bruns2016Quant-52322, title={Quantum Jumps of Normal Polytopes}, year={2016}, doi={10.1007/s00454-016-9773-7}, number={1}, volume={56}, issn={0179-5376}, journal={Discrete & Computational Geometry}, pages={181--215}, author={Bruns, Winfried and Gubeladze, Joseph and Michalek, Mateusz} }

Gubeladze, Joseph 2021-01-08T12:21:34Z Gubeladze, Joseph 2016 eng Bruns, Winfried Bruns, Winfried 2021-01-08T12:21:34Z We introduce a partial order on the set of all normal polytopes in R<sup>d</sup>. This poset NPol(d) is a natural discrete counterpart of the continuum of convex compact sets in R<sup>d</sup>, 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. Quantum Jumps of Normal Polytopes Michalek, Mateusz Michalek, Mateusz terms-of-use

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