Computational determination of the largest lattice polytope diameter

A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let δ(d,k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational framework to determine δ(d,k) for small instances. We show that δ(3,4)=7 and δ(3,5)=9; that is, we verify for (d,k)=(3,4) and (3, 5) the conjecture whereby δ(d,k) is at most ⌊(k+1)d/2⌋ and is achieved, up to translation, by a Minkowski sum of lattice vectors.