On the maximal width of empty lattice simplices (Q1964652)

A \(k\)-dimensional lattice simplex \(S\subset\mathbb{R}^d\) is the convex hull of \(k+1\) affinely independent points from \(\mathbb{Z}^d\), and \(S\) is said to be empty if it intersects \(\mathbb{Z}^d\) only at its vertices. Unexpectedly, the authors construct infinitely many empty lattice \(d\)-simplices of arbitrarily large volume from \((d-1)\)-dimensional ones of the same lattice width, yielding in particular also an infinite sequence of empty lattice 4-simplices of width 2. Modifying a former conjecture they formulate the following finiteness conjecture: For every \(d\geq 2\), there are only finitely many equivalence classes of empty \(d\)-simplices whose width is larger than the greatest width that can be achieved by almost empty simplices in dimension \(d-1\). (A lattice simplex without interior lattice points and with at least one empty facet is called almost empty).