Improved bounds on the diameter of lattice polytopes

We show that the largest possible diameter ({delta(d,k)}) of a d-dimensional polytope whose vertices have integer coordinates ranging between 0 and k is at most ({kd - lceil2d/3rceil-(k-3)}) when ({kgeq3}) . In addition, we show that ({delta(4,3)=8}) . This substantiates the conjecture whereby ({delta(d,k)}) is at most ({lfloor(k+1)d/2rfloor}) and is achieved by a Minkowski sum of lattice vectors.