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.