On the Number of Lattice Free Polytopes

AbstractV. I. Arnold asked for estimates for the number of equivalence classes of lattice polytopes, under the group of unimodular affine transformations. What we investigate here is the analogous question for lattice free polytopes. Some of the results: the number of equivalence classes of lattice free simplices of volume at most v in dimension d is of order vd−1, and the number of equivalence classes of lattice free polytopes of volume at most v in dimension d isO (v2d−1(logv )d−2)

On Lattice-Free Orbit Polytopes

Given a permutation group acting on coordinates of $\mathbb{R}^n$, we consider lattice-free polytopes that are the convex hull of an orbit of one integral vector. The vertices of such polytopes are called \emph{core points} and they play a key role in a recent approach to exploit symmetry in integer convex optimization problems. Here, naturally the question arises, for which groups the number of core points is finite up to translations by vectors fixed by the group. In this paper we consider transitive permutation groups and prove this type of finiteness for the $2$-homogeneous ones. We provide tools for practical computations of core points and obtain a complete list of representatives for all $2$-homogeneous groups up to degree twelve. For transitive groups that are not $2$-homogeneous we conjecture that there exist infinitely many core points up to translations by the all-ones-vector. We prove our conjecture for two large classes of groups: For imprimitive groups and groups that have an irrational invariant subspace.Comment: 27 pages, 2 figures; with minor adaptions according to referee comments; to appear in Discrete and Computational Geometr