Improved bounds on the diameter of lattice polytopes

We show that the largest possible diameter δ(d,k)$${\delta(d,k)}$$ of a d-dimensional polytope whose vertices have integer coordinates ranging between 0 and k is at most kd-⌈2d/3⌉-(k-3)$${kd - \lceil2d/3\rceil-(k-3)}$$ when k≥3$${k\geq3}$$ . In addition, we show that δ(4,3)=8$${\delta(4,3)=8}$$ . This substantiates the conjecture whereby δ(d,k)$${\delta(d,k)}$$ is at most ⌊(k+1)d/2⌋$${\lfloor(k+1)d/2\rfloor}$$ and is achieved by a Minkowski sum of lattice vectors.