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Largest integral simplices with one interior integral point - solution of Hensley's conjecture and related results

  • For each dimension d, d-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex with precisely one interior integral point, we give bounds on the volumes of its faces, the barycentric coordinates of the interior integral point and its number of integral points. Furthermore, we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points. The presented results have applications in toric geometry and in integer optimization.

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Metadaten
Author: Gennadiy Averkov, Jan Krümpelmann, Benjamin Nill
URL:https://www.sciencedirect.com/science/article/pii/S0001870815000249?via%3Dihub#!
DOI:https://doi.org/10.1016/j.aim.2014.12.035
ISSN:0001-8708
Title of the source (English):Advances in Mathematic
Document Type:Scientific journal article peer-reviewed
Language:English
Year of publication:2015
Volume/Year:Volume 274
First Page:118
Last Page:166
Faculty/Chair:Fakultät 1 MINT - Mathematik, Informatik, Physik, Elektro- und Informationstechnik / FG Algorithmische Mathematik
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