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Largest integral simplices with one interior integral point: Solution of Hensley's conjecture and related results | | Gennadiy Averkov
; Jan Krümpelmann
; Benjamin Nill
; | | Date: |
30 Sep 2013 | | Abstract: | 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, and
Pikhurko. 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, its
barycentric coordinates 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. | | Source: | arXiv, 1309.7967 | | Services: | Forum | Review | PDF | Favorites | |
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