2011/03/03 by Averkov, Gennadiy
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1103.0629
Let Td(1) be the set of all d-dimensional simplices T in \reald with integer vertices and a single integer point in the interior of T. It follows from a result of Hensley that Td(1) is finite up to affine transformations that preserve ℤd. It is known that, when d grows, the maximum volume of the simplices T ∈ \cTd(1) becomes extremely large. We improve and refine bounds on the size of T ∈ Td(1) (where by the size we mean the volume or the number of lattice points). It is shown that each T ∈ Td(1) can be decomposed into an ascending chain of faces whose sizes are `not too large'. More precisely, if T ∈ Td(1), then there exist faces G1 ⊆ ... ⊆ Gd=T of T such that, for every i ∈ \1,...,d\, Gi is i-dimensional and the size of Gi is bounded from above in terms of i and d. The bound on the size of Gi is double exponential in i. The presented upper bounds are asymptotically tight on the log-log scale.