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 $kd-\lceil 2d\mathrm{/}3\rceil$ when $k\ge 3$. In addition, we show that $\delta (4,3)=8$. This substantiates the conjecture whereby $\delta (d,k)$ is at most $\lfloor (k+1)d\mathrm{/}2\rfloor$ and is achieved by a Minkowski sum of lattice vectors.