Quantum Jumps of Normal Polytopes
Autor(en): | Bruns, Winfried Gubeladze, Joseph Michalek, Mateusz |
Stichwörter: | Computer Science; Computer Science, Theory & Methods; INTEGER ANALOG; Lattice polytope; Mathematics; Maximal polytope; Normal polytope; Quantum jump | Erscheinungsdatum: | 2016 | Herausgeber: | SPRINGER | Journal: | DISCRETE & COMPUTATIONAL GEOMETRY | Volumen: | 56 | Ausgabe: | 1 | Startseite: | 181 | Seitenende: | 215 | Zusammenfassung: | We introduce a partial order on the set of all normal polytopes in . This poset is a natural discrete counterpart of the continuum of convex compact sets in , ordered by inclusion, and exhibits a remarkably rich combinatorial structure. We derive various arithmetic bounds on elementary relations in , called quantum jumps. The existence of extremal objects in is a challenge of number theoretical flavor, leading to interesting classes of normal polytopes: minimal, maximal, spherical. Minimal elements in 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 and , found by a combination of the developed theory, random generation, and extensive computer search. |
ISSN: | 01795376 | DOI: | 10.1007/s00454-016-9773-7 |
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