Lattice points in lattice polytopes

Abstract

It is shown that, for any lattice polytope PaW 1 , the set int {P)nlZ d (provided that it is non-empty) contains a point whose coefficient of asymmetry with respect to P is at most Sd-(8/+1) 2 ' * . If, moreover, P is a simplex, then this bound can be improved to 8 ( 8 / + I ) 2 . As an application, new upper bounds on the volume of a lattice polytope are deduced, given its dimension and the number of sublattice points in its interior. §1. Introduction. A lattice polytope in W 1 is a convex polytope whose vertices are lattice points, that is, points in ~L d . For an integer /2=1, let

Maximal Lattice-Free Polyhedra: Finiteness and an Explicit Description in Dimension Three

“…Furthermore, int(Q) ∩ Λ = ∅ since P is properly contained in Q and P is maximal Λ-free. 2d+1 (see [Pik01]). …”

Fano polytopes

“…If P is canonical, Pikhurko [46] gives an upper bound on the sum of the weights h  2 3n 2 15 (n 1)2 n+1 . In dimensions two and three this is far from sharp (the maximum values are 1 + 2 + 3 = 6 and 5 + 6 + 22 + 33 = 66, respectively).…”

Volume and Lattice Points of Reflexive Simplices

“…Convex and discrete geometry: Given a d-dimensional lattice polytope P with only one lattice point in its interior, Hensley [13] showed that the volume and the number of lattice points of P is bounded above by a function depending only on d. In [21] and [25] asymptotically better upper bounds were obtained. However, in lower dimensions they are presumably still very far from being sharp, cf.…”

“…This bound vastly improves on general upper bounds on the volume of lattice simplices containing only one lattice point in the interior, as given in [21] or [25].…”