2013/09/30 by Gennadiy Averkov, Averkov, Gennadiy, Jan Krümpelmann +3
Mathematics · #14M25 #52B20 #90C11 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1309.7967
openalex publication_date 2013/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For each dimension d, d-dimensional integral simplices with exactly one\ninterior integral point have bounded volume. This was first shown by Hensley.\nExplicit volume bounds were determined by Hensley, Lagarias and Ziegler,\nPikhurko, and Averkov. In this paper we determine the exact upper volume bound\nfor such simplices and characterize the volume-maximizing simplices. We also\ndetermine the sharp upper bound on the coefficient of asymmetry of an integral\npolytope with a single interior integral point. This result confirms a\nconjecture of Hensley from 1983. Moreover, for an integral simplex with\nprecisely one interior integral point, we give bounds on the volumes of its\nfaces, the barycentric coordinates of the interior integral point and its\nnumber of integral points. Furthermore, we prove a bound on the lattice\ndiameter of integral polytopes with a fixed number of interior integral points.\nThe presented results have applications in toric geometry and in integer\noptimization.\n