Quantum jumps of normal polytopes
Abstract
We introduce a partial order on the set of all normal polytopes in R^d. This poset NPol(d) is a natural discrete counterpart of the continuum of convex compact sets in R^d, 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.
Cite
@article{arxiv.1504.01036,
title = {Quantum jumps of normal polytopes},
author = {Winfried Bruns and Joseph Gubeladze and Mateusz Michałek},
journal= {arXiv preprint arXiv:1504.01036},
year = {2016}
}
Comments
Final version, to appear in Discrete & Computational Geometry