On the Maximal Width of Empty Lattice Simplices

A k-dimensional lattice simplex σ ⊆ Rd is the convex hull of k + 1 affinely independent integer points. General lattice polytopes are obtained by taking convex hulls of arbitrary finite subsets of Zd . A lattice simplex or polytope is called empty if it intersects the lattice Zd only at its vertices. (Such polytopes are studied also under the names elementary and latticefree.) In dimensions d > 3, not all empty lattice simplices have lattice width 1 (that is, they are not all enclosed by two adjacent lattice hyperplanes). However, the famous ‘flatness theorem’ of Khinchine (see [1, 2]) implies that the maximal width of an empty lattice simplex is bounded by a constant w(d) in each fixed dimension d . Additionally, Bárány conjectured (personal communication) that in each dimension the volume of an empty lattice simplex of width greater than 1 is bounded (equivalently, there are only finitely many combinatorial types, up to unimodular equivalence). In this paper, we disprove Bárány’s conjecture. Even stronger, we show that for every ‘almost empty’ lattice simplex of dimension d , there are infinitely many empty lattice simplices of the same width in dimension d + 1. In particular, this produces an infinite sequence of lattice simplices of width 2 in dimension 4. We also propose a modified finiteness conjecture and present some (computational) evidence for it. In the following, we consider lattice simplices only up to unimodular transformations. Thus, examples of lattice simplices are considered to be different if they cannot be related by a lattice-preserving affine map. The determinant of a d-dimensional lattice simplex σ = conv{a0, a1, . . . , ad} ⊂ Rd is given by det(σ ) := | det(a1 − a0, . . . , ad − a0)|. The volume of σ is then 1 d! det(σ ). Let K ⊆ Rd be any full-dimensional lattice polytope (or even a general full-dimensional convex body). For a linear form ` ∈ (Rd)∗ define the width of K with respect to ` as